ldquoBaseball is 90 mental The other half is physicalrdquo

Yogi Berra

Probabilitybull Denoted by P(Event)

outcomes total

outcomes favorable)( EP

This method for calculating probabilities is only appropriate when the outcomes of the sample space are equally likely

Law of Large Numbers

bull As the number of repetitions of a chance experiment increase the difference between the relative frequency of occurrence for an event and the true probability approaches zero

Basic Rules of ProbabilityRule 1 Legitimate Values

For any event E 0 lt P(E) lt 1

A probabilityis a number between 0 and 1

The probability of rain must be 110

Rule 2 Sample spaceIf S is the sample space P(S) = 1

ldquoSomething Has to Happen Rulerdquo

The probability of the set of all possible outcomes must be 1

Irsquom 100 sure you are going to have a boyhellip or a girl

If the probability that you get to class on time is 8 then the probability that you do not get to class on time is 2

Rule 3 ComplementFor any event E

P(E) + P(not E) = 1 P(E) = 1 ndash P(not E)

Independent

bull Two events are independent if knowing that one will occur (or has occurred) does not change the probability that the other occursndash A randomly selected student is female - What is

the probability she plays soccer for SHS

ndash A randomly selected student is female - What is the probability she plays football for SHS

Independent

Dependent

Rule 4 Multiplication

If two events A amp B are independent

General rule

P(B) P(A) B) ampP(A

A)|P(B P(A) B) ampP(A If the probability of rolling a 5 on a fair dice is 16 what is the probability of rolling a 5 three times in a row

P( three 5rsquos in a row) = (16) x (16) x (16) = 1216 or 004629

)( BAP

Independent)( BAP

Yes

)()( BPAP

What does this mean

Given a deck of cards and a die

one card is drawn and the dice is rolled

What is the probability that an ace is drawn and an even is

rolledP(ace and even) = P(ace) P(even) =

4 3 1

52 6 26

)( BAP

Independent)( BAP

Yes No

)()( BPAP )|()( ABPAP

Given a deck of cards two

cards are drawn

without replacement What is the probability

that they are both heartsP(heart and heart) =

)|()( ABPAP

13 12 1

52 51 17

Ex 6) Suppose I will pick two cards from a standard deck without replacement What is the probability that I select two spades

Are the cards independent NO

P(A amp B) = P(A) P(B|A)

Read ldquoprobability of B given that A occursrdquo

P(Spade amp Spade) = 14 1251 = 117

The probability of getting a spade given that a spade has already been drawn

Rule 5 AdditionIf two events E amp F are disjoint

P(E or F) = P(E) + P(F)

If the probability that a randomly selected student is a junior (A) is 2 and the probability that the student is a senior (B) is 5 what is the probability that the student is either a junior or a senior

P(A υ B) = P(A) + P(B) if A and B are disjoint

P(A υ B) = 2 + 5 = 7

bull Two events that have no common outcomes are said to be disjoint or mutually exclusive

A and B are disjoint events

Rule 5 AdditionIf two events E amp F are disjoint P(E or F) = P(E) + P(F)

(General) If two events E amp F are not disjoint

P(E or F) = P(E) + P(F) ndash P(E amp F)Probability of owning a MP3 player 50Probability of owning a computer 90

So the probability of owning a MP3 player or a computer

is 140

Not disjoint events

)( FEP What does this mean

)( FEP Mutually exclusive

)()( FPEP

Yes

Given a deck of cards one

card is drawn What

is the probability

that it is a 3 or a 4

P(3 or 4) = P(3) + P(4) =

4 4 2

52 52 13

)( FEP

)( FEP Mutually exclusive

)()( FPEP

Yes No

)()()( FEPFPEP

Given a deck of cards one

card is drawn What

is the probability that it is an ace or a red

cardP(ace or red) = P(ace) + P(red) ndash P(ace and red) =

52

4

52

26

52

2

52

28+ - =

)( FEP

)( FEP Mutually exclusive

)()( FPEP

Yes No

)()()( FEPFPEP

Independent Yes )()( FPEP

Ex 5)

If P(A) = 045 P(B) = 035 and A amp B are independent find P(A or B)Is A amp B disjoint

If A amp B are disjoint are they independent

Disjoint events do not happen at the same time

So if A occurs can B occur

Disjoint events are dependent

NO independent events cannot be disjoint

P(A or B) = P(A) + P(B) ndash P(A amp B)

How can you find the

probability of A amp B

P(A or B) = 45 + 35 - 45(35) = 06425

If independent

multiply

In a class there are 12 boys made up of 8 Seniors and 4 Juniors There are also 8 girls made up of 3 Seniors and 5 Juniors Find the probability of choosing a boy or a Senior

Note that choosing a boy and choosing a Senior are not disjoint (they can occur simultaneously)

P(boy or a senior) = P(Boy) + P(Senior) ndash P(Senior boy) = 12 11 8 15 3

20 20 20 20 4

Boy Girl Total

Senior 8 3 11

Junior 4 5 9

Total 12 8 20

Rule 6 At least one

The probability that at least one outcome happens is 1 minus the probability that no outcomes happen

P(at least 1) = 1 ndash P(none)

For a sales promotion the manufacturer places winning symbols under the caps of 10 of all Dr Pepper bottles You buy a six-pack What is the probability that you win something

P(at least one winning symbol) =

1 ndash P(no winning symbols) 1 - 96 = 4686

Suppose that 40 of cars in Fort Smith are manufactured in the United States 30 in Japan 10 in Germany and 20 in other countries

If cars are selected at random what is the probability that it is not US made

P(not US made) = 1 ndash P(US made) = 1 - 4 = 6

Suppose that 40 of cars in Fort Smith are manufactured in the United States 30 in Japan 10 in Germany and 20 in other countries

If cars are selected at random what is the probability that it is made in Japan or Germany

P(Japanese or German) = P(Japanese) + P(German) = 3 + 1 = 4

Suppose that 40 of cars in Fort Smith are manufactured in the United States 30 in Japan 10 in Germany and 20 in other countries

If cars are selected at random what is the probability that you see two in a row from Japan

P(2 Japanese in a row) = P(Japanese) and P(Japanese) = P(J) x P(J) = 3 x 3 = 09

Suppose that 40 of cars in Fort Smith are manufactured in the United States 30 in Japan 10 in Germany and 20 in other countries

If cars are selected at random what is the probability that none of three cars came from Germany

P(no Germany in three) = P(not G) x P(not G) x P(not G) = 9 x 9 x 9 = 729

Suppose that 40 of cars in Fort Smith are manufactured in the United States 30 in Japan 10 in Germany and 20 in other countries

If cars are selected at random what is the probability that at least one of three cars is US made

P(at least one US in three) = 1 ndash P(no US in three) = 1 ndash (6)(6)(6) = 784

Suppose that 40 of cars in Fort Smith are manufactured in the United States 30 in Japan 10 in Germany and 20 in other countries

If cars are selected at random what is the probability that the first Japanese car is the fourth one you choose

P(first J is the fourth car) = P(not J) x P(not J) x P(not J) x P(J) = (7)3 (3) = 1029

Watch out for

bull probabilities that donrsquot add up to 1

bull donrsquot add probabilities of events if they are not disjoint

bull donrsquot multiply probabilities of events if they are not independent

bull donrsquot confuse disjoint and independent

Rule 7 Conditional Probability

bull A probability that takes into account a given condition

P(A)

B)P(AA)|P(B

P(given)

P(and)A)|P(B

In a class there are 12 boys made up of 8 Seniors and 4 Juniors There are also 8 girls made up of 3 Seniors and 5 Juniors Find the probability of choosing a boy given that he is a Senior

Boy Girl Total

Senior 8 3 11

Junior 4 5 9

Total 12 8 20

P(Boy|Senior) = 811

P(Senior|Boy) = 812 = 23

Probabilities from two way tables

Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359

12) What is the probability that the driver is a student

359195

)( StudentP

Probabilities from two way tables

Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359

13) What is the probability that the driver drives a European car

35945

)( EuropeanP

Probabilities from two way tables

Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359

14) What is the probability that the driver drives an American or Asian car

Disjoint359102212

)(

AsianorAmericanP

Probabilities from two way tables

Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359

15) What is the probability that the driver is staff or drives an Asian car

Disjoint35947102164

)(

AsianorStaffP

Probabilities from two way tables

Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359

16) What is the probability that the driver is staff and drives an Asian car

35947

)( AsianandStaffP

Probabilities from two way tables

Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359

17) If the driver is a student what is the probability that they drive an American car

Condition195107

)|( StudentAmericanP

Probabilities from two way tables

Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359

18) What is the probability that the driver is a student if the driver drives a European car

Condition 4533

)|( EuropeanStudentP

Definition of Independent Events

Two events E and F are independent if and only if

P(F | E) = P(F) or P(E | F) = P(E)

EXAMPLE Illustrating Independent Events

The probability a randomly selected murder victim is male is 07515 The probability a randomly selected murder victim is male given that they are less than 18 years old is 06751

Since P(male) = 07515 and

P(male | lt 18 years old) = 06751

the events ldquomalerdquo and ldquoless than 18 years oldrdquo are not independent In fact knowing the victim is less than 18 years old decreases the probability that the victim is male

I draw one card and look

at it I tell you it is red What is the

probability it is a heart

P( heart | red) =

13P(heart and red) 152

26P(red) 252

Are ldquored cardrdquo and ldquospaderdquo mutually

exclusive Are they

independentA red card canrsquot be a spade so they ARE mutually exclusive

Are ldquored cardrdquo and

ldquoacerdquo mutually

exclusive Are they

independent2 aces are

red cards so they are

NOT mutually exclusive

Are ldquoface cardrdquo and

ldquokingrdquo mutually

exclusive Are they

independentKings are

Face cards so they are

NOT mutually exclusive

ldquoSlump I ainrsquot in no slump I just ainrsquot hittinrdquo

Yogi Berra

• PowerPoint Presentation
• Probability
• Law of Large Numbers
• Basic Rules of Probability
• Slide 5
• Slide 6
• Independent
• Slide 8
• Slide 9
• Slide 10
• Slide 11
• Slide 12
• Slide 13
• Slide 14
• Slide 15
• Slide 16
• Slide 17
• Slide 18
• Slide 19
• Slide 20
• Slide 21
• Slide 22
• Slide 23
• Slide 24
• Slide 25
• Slide 26
• Slide 27
• Slide 28
• Slide 29
• Slide 30
• Slide 31
• Slide 32
• Rule 7 Conditional Probability
• Slide 34
• Probabilities from two way tables
• Slide 36
• Slide 37
• Slide 38
• Slide 39
• Slide 40
• Slide 41
• Slide 42
• Slide 43
• Slide 44
• Slide 45
• Slide 46
• Slide 47
• Slide 48

Probabilitybull Denoted by P(Event)

outcomes total

outcomes favorable)( EP

This method for calculating probabilities is only appropriate when the outcomes of the sample space are equally likely

Law of Large Numbers

bull As the number of repetitions of a chance experiment increase the difference between the relative frequency of occurrence for an event and the true probability approaches zero

Basic Rules of ProbabilityRule 1 Legitimate Values

For any event E 0 lt P(E) lt 1

A probabilityis a number between 0 and 1

The probability of rain must be 110

Rule 2 Sample spaceIf S is the sample space P(S) = 1

ldquoSomething Has to Happen Rulerdquo

The probability of the set of all possible outcomes must be 1

Irsquom 100 sure you are going to have a boyhellip or a girl

If the probability that you get to class on time is 8 then the probability that you do not get to class on time is 2

Rule 3 ComplementFor any event E

P(E) + P(not E) = 1 P(E) = 1 ndash P(not E)

Independent

bull Two events are independent if knowing that one will occur (or has occurred) does not change the probability that the other occursndash A randomly selected student is female - What is

the probability she plays soccer for SHS

ndash A randomly selected student is female - What is the probability she plays football for SHS

Independent

Dependent

Rule 4 Multiplication

If two events A amp B are independent

General rule

P(B) P(A) B) ampP(A

A)|P(B P(A) B) ampP(A If the probability of rolling a 5 on a fair dice is 16 what is the probability of rolling a 5 three times in a row

P( three 5rsquos in a row) = (16) x (16) x (16) = 1216 or 004629

)( BAP

Independent)( BAP

Yes

)()( BPAP

What does this mean

Given a deck of cards and a die

one card is drawn and the dice is rolled

What is the probability that an ace is drawn and an even is

rolledP(ace and even) = P(ace) P(even) =

4 3 1

52 6 26

)( BAP

Independent)( BAP

Yes No

)()( BPAP )|()( ABPAP

Given a deck of cards two

cards are drawn

without replacement What is the probability

that they are both heartsP(heart and heart) =

)|()( ABPAP

13 12 1

52 51 17

Ex 6) Suppose I will pick two cards from a standard deck without replacement What is the probability that I select two spades

Are the cards independent NO

P(A amp B) = P(A) P(B|A)

Read ldquoprobability of B given that A occursrdquo

P(Spade amp Spade) = 14 1251 = 117

The probability of getting a spade given that a spade has already been drawn

Rule 5 AdditionIf two events E amp F are disjoint

P(E or F) = P(E) + P(F)

If the probability that a randomly selected student is a junior (A) is 2 and the probability that the student is a senior (B) is 5 what is the probability that the student is either a junior or a senior

P(A υ B) = P(A) + P(B) if A and B are disjoint

P(A υ B) = 2 + 5 = 7

bull Two events that have no common outcomes are said to be disjoint or mutually exclusive

A and B are disjoint events

Rule 5 AdditionIf two events E amp F are disjoint P(E or F) = P(E) + P(F)

(General) If two events E amp F are not disjoint

P(E or F) = P(E) + P(F) ndash P(E amp F)Probability of owning a MP3 player 50Probability of owning a computer 90

So the probability of owning a MP3 player or a computer

is 140

Not disjoint events

)( FEP What does this mean

)( FEP Mutually exclusive

)()( FPEP

Yes

Given a deck of cards one

card is drawn What

is the probability

that it is a 3 or a 4

P(3 or 4) = P(3) + P(4) =

4 4 2

52 52 13

)( FEP

)( FEP Mutually exclusive

)()( FPEP

Yes No

)()()( FEPFPEP

Given a deck of cards one

card is drawn What

is the probability that it is an ace or a red

cardP(ace or red) = P(ace) + P(red) ndash P(ace and red) =

52

4

52

26

52

2

52

28+ - =

)( FEP

)( FEP Mutually exclusive

)()( FPEP

Yes No

)()()( FEPFPEP

Independent Yes )()( FPEP

Ex 5)

If P(A) = 045 P(B) = 035 and A amp B are independent find P(A or B)Is A amp B disjoint

If A amp B are disjoint are they independent

Disjoint events do not happen at the same time

So if A occurs can B occur

Disjoint events are dependent

NO independent events cannot be disjoint

P(A or B) = P(A) + P(B) ndash P(A amp B)

How can you find the

probability of A amp B

P(A or B) = 45 + 35 - 45(35) = 06425

If independent

multiply

In a class there are 12 boys made up of 8 Seniors and 4 Juniors There are also 8 girls made up of 3 Seniors and 5 Juniors Find the probability of choosing a boy or a Senior

Note that choosing a boy and choosing a Senior are not disjoint (they can occur simultaneously)

P(boy or a senior) = P(Boy) + P(Senior) ndash P(Senior boy) = 12 11 8 15 3

20 20 20 20 4

Boy Girl Total

Senior 8 3 11

Junior 4 5 9

Total 12 8 20

Rule 6 At least one

The probability that at least one outcome happens is 1 minus the probability that no outcomes happen

P(at least 1) = 1 ndash P(none)

For a sales promotion the manufacturer places winning symbols under the caps of 10 of all Dr Pepper bottles You buy a six-pack What is the probability that you win something

P(at least one winning symbol) =

1 ndash P(no winning symbols) 1 - 96 = 4686

Suppose that 40 of cars in Fort Smith are manufactured in the United States 30 in Japan 10 in Germany and 20 in other countries

If cars are selected at random what is the probability that it is not US made

P(not US made) = 1 ndash P(US made) = 1 - 4 = 6

Suppose that 40 of cars in Fort Smith are manufactured in the United States 30 in Japan 10 in Germany and 20 in other countries

If cars are selected at random what is the probability that it is made in Japan or Germany

P(Japanese or German) = P(Japanese) + P(German) = 3 + 1 = 4

Suppose that 40 of cars in Fort Smith are manufactured in the United States 30 in Japan 10 in Germany and 20 in other countries

If cars are selected at random what is the probability that you see two in a row from Japan

P(2 Japanese in a row) = P(Japanese) and P(Japanese) = P(J) x P(J) = 3 x 3 = 09

Suppose that 40 of cars in Fort Smith are manufactured in the United States 30 in Japan 10 in Germany and 20 in other countries

If cars are selected at random what is the probability that none of three cars came from Germany

P(no Germany in three) = P(not G) x P(not G) x P(not G) = 9 x 9 x 9 = 729

Suppose that 40 of cars in Fort Smith are manufactured in the United States 30 in Japan 10 in Germany and 20 in other countries

If cars are selected at random what is the probability that at least one of three cars is US made

P(at least one US in three) = 1 ndash P(no US in three) = 1 ndash (6)(6)(6) = 784

Suppose that 40 of cars in Fort Smith are manufactured in the United States 30 in Japan 10 in Germany and 20 in other countries

If cars are selected at random what is the probability that the first Japanese car is the fourth one you choose

P(first J is the fourth car) = P(not J) x P(not J) x P(not J) x P(J) = (7)3 (3) = 1029

Watch out for

bull probabilities that donrsquot add up to 1

bull donrsquot add probabilities of events if they are not disjoint

bull donrsquot multiply probabilities of events if they are not independent

bull donrsquot confuse disjoint and independent

Rule 7 Conditional Probability

bull A probability that takes into account a given condition

P(A)

B)P(AA)|P(B

P(given)

P(and)A)|P(B

In a class there are 12 boys made up of 8 Seniors and 4 Juniors There are also 8 girls made up of 3 Seniors and 5 Juniors Find the probability of choosing a boy given that he is a Senior

Boy Girl Total

Senior 8 3 11

Junior 4 5 9

Total 12 8 20

P(Boy|Senior) = 811

P(Senior|Boy) = 812 = 23

Probabilities from two way tables

Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359

12) What is the probability that the driver is a student

359195

)( StudentP

Probabilities from two way tables

Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359

13) What is the probability that the driver drives a European car

35945

)( EuropeanP

Probabilities from two way tables

Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359

14) What is the probability that the driver drives an American or Asian car

Disjoint359102212

)(

AsianorAmericanP

Probabilities from two way tables

Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359

15) What is the probability that the driver is staff or drives an Asian car

Disjoint35947102164

)(

AsianorStaffP

Probabilities from two way tables

Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359

16) What is the probability that the driver is staff and drives an Asian car

35947

)( AsianandStaffP

Probabilities from two way tables

Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359

17) If the driver is a student what is the probability that they drive an American car

Condition195107

)|( StudentAmericanP

Probabilities from two way tables

Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359

18) What is the probability that the driver is a student if the driver drives a European car

Condition 4533

)|( EuropeanStudentP

Definition of Independent Events

Two events E and F are independent if and only if

P(F | E) = P(F) or P(E | F) = P(E)

EXAMPLE Illustrating Independent Events

The probability a randomly selected murder victim is male is 07515 The probability a randomly selected murder victim is male given that they are less than 18 years old is 06751

Since P(male) = 07515 and

P(male | lt 18 years old) = 06751

the events ldquomalerdquo and ldquoless than 18 years oldrdquo are not independent In fact knowing the victim is less than 18 years old decreases the probability that the victim is male

I draw one card and look

at it I tell you it is red What is the

probability it is a heart

P( heart | red) =

13P(heart and red) 152

26P(red) 252

Are ldquored cardrdquo and ldquospaderdquo mutually

exclusive Are they

independentA red card canrsquot be a spade so they ARE mutually exclusive

Are ldquored cardrdquo and

ldquoacerdquo mutually

exclusive Are they

independent2 aces are

red cards so they are

NOT mutually exclusive

Are ldquoface cardrdquo and

ldquokingrdquo mutually

exclusive Are they

independentKings are

Face cards so they are

NOT mutually exclusive

ldquoSlump I ainrsquot in no slump I just ainrsquot hittinrdquo

Yogi Berra

• PowerPoint Presentation
• Probability
• Law of Large Numbers
• Basic Rules of Probability
• Slide 5
• Slide 6
• Independent
• Slide 8
• Slide 9
• Slide 10
• Slide 11
• Slide 12
• Slide 13
• Slide 14
• Slide 15
• Slide 16
• Slide 17
• Slide 18
• Slide 19
• Slide 20
• Slide 21
• Slide 22
• Slide 23
• Slide 24
• Slide 25
• Slide 26
• Slide 27
• Slide 28
• Slide 29
• Slide 30
• Slide 31
• Slide 32
• Rule 7 Conditional Probability
• Slide 34
• Probabilities from two way tables
• Slide 36
• Slide 37
• Slide 38
• Slide 39
• Slide 40
• Slide 41
• Slide 42
• Slide 43
• Slide 44
• Slide 45
• Slide 46
• Slide 47
• Slide 48

Law of Large Numbers

bull As the number of repetitions of a chance experiment increase the difference between the relative frequency of occurrence for an event and the true probability approaches zero

Basic Rules of ProbabilityRule 1 Legitimate Values

For any event E 0 lt P(E) lt 1

A probabilityis a number between 0 and 1

The probability of rain must be 110

Rule 2 Sample spaceIf S is the sample space P(S) = 1

ldquoSomething Has to Happen Rulerdquo

The probability of the set of all possible outcomes must be 1

Irsquom 100 sure you are going to have a boyhellip or a girl

If the probability that you get to class on time is 8 then the probability that you do not get to class on time is 2

Rule 3 ComplementFor any event E

P(E) + P(not E) = 1 P(E) = 1 ndash P(not E)

Independent

bull Two events are independent if knowing that one will occur (or has occurred) does not change the probability that the other occursndash A randomly selected student is female - What is

the probability she plays soccer for SHS

ndash A randomly selected student is female - What is the probability she plays football for SHS

Independent

Dependent

Rule 4 Multiplication

If two events A amp B are independent

General rule

P(B) P(A) B) ampP(A

A)|P(B P(A) B) ampP(A If the probability of rolling a 5 on a fair dice is 16 what is the probability of rolling a 5 three times in a row

P( three 5rsquos in a row) = (16) x (16) x (16) = 1216 or 004629

)( BAP

Independent)( BAP

Yes

)()( BPAP

What does this mean

Given a deck of cards and a die

one card is drawn and the dice is rolled

What is the probability that an ace is drawn and an even is

rolledP(ace and even) = P(ace) P(even) =

4 3 1

52 6 26

)( BAP

Independent)( BAP

Yes No

)()( BPAP )|()( ABPAP

Given a deck of cards two

cards are drawn

without replacement What is the probability

that they are both heartsP(heart and heart) =

)|()( ABPAP

13 12 1

52 51 17

Ex 6) Suppose I will pick two cards from a standard deck without replacement What is the probability that I select two spades

Are the cards independent NO

P(A amp B) = P(A) P(B|A)

Read ldquoprobability of B given that A occursrdquo

P(Spade amp Spade) = 14 1251 = 117

The probability of getting a spade given that a spade has already been drawn

Rule 5 AdditionIf two events E amp F are disjoint

P(E or F) = P(E) + P(F)

If the probability that a randomly selected student is a junior (A) is 2 and the probability that the student is a senior (B) is 5 what is the probability that the student is either a junior or a senior

P(A υ B) = P(A) + P(B) if A and B are disjoint

P(A υ B) = 2 + 5 = 7

bull Two events that have no common outcomes are said to be disjoint or mutually exclusive

A and B are disjoint events

Rule 5 AdditionIf two events E amp F are disjoint P(E or F) = P(E) + P(F)

(General) If two events E amp F are not disjoint

P(E or F) = P(E) + P(F) ndash P(E amp F)Probability of owning a MP3 player 50Probability of owning a computer 90

So the probability of owning a MP3 player or a computer

is 140

Not disjoint events

)( FEP What does this mean

)( FEP Mutually exclusive

)()( FPEP

Yes

Given a deck of cards one

card is drawn What

is the probability

that it is a 3 or a 4

P(3 or 4) = P(3) + P(4) =

4 4 2

52 52 13

)( FEP

)( FEP Mutually exclusive

)()( FPEP

Yes No

)()()( FEPFPEP

Given a deck of cards one

card is drawn What

is the probability that it is an ace or a red

cardP(ace or red) = P(ace) + P(red) ndash P(ace and red) =

52

4

52

26

52

2

52

28+ - =

)( FEP

)( FEP Mutually exclusive

)()( FPEP

Yes No

)()()( FEPFPEP

Independent Yes )()( FPEP

Ex 5)

If P(A) = 045 P(B) = 035 and A amp B are independent find P(A or B)Is A amp B disjoint

If A amp B are disjoint are they independent

Disjoint events do not happen at the same time

So if A occurs can B occur

Disjoint events are dependent

NO independent events cannot be disjoint

P(A or B) = P(A) + P(B) ndash P(A amp B)

How can you find the

probability of A amp B

P(A or B) = 45 + 35 - 45(35) = 06425

If independent

multiply

In a class there are 12 boys made up of 8 Seniors and 4 Juniors There are also 8 girls made up of 3 Seniors and 5 Juniors Find the probability of choosing a boy or a Senior

Note that choosing a boy and choosing a Senior are not disjoint (they can occur simultaneously)

P(boy or a senior) = P(Boy) + P(Senior) ndash P(Senior boy) = 12 11 8 15 3

20 20 20 20 4

Boy Girl Total

Senior 8 3 11

Junior 4 5 9

Total 12 8 20

Rule 6 At least one

The probability that at least one outcome happens is 1 minus the probability that no outcomes happen

P(at least 1) = 1 ndash P(none)

For a sales promotion the manufacturer places winning symbols under the caps of 10 of all Dr Pepper bottles You buy a six-pack What is the probability that you win something

P(at least one winning symbol) =

1 ndash P(no winning symbols) 1 - 96 = 4686

Suppose that 40 of cars in Fort Smith are manufactured in the United States 30 in Japan 10 in Germany and 20 in other countries

If cars are selected at random what is the probability that it is not US made

P(not US made) = 1 ndash P(US made) = 1 - 4 = 6

Suppose that 40 of cars in Fort Smith are manufactured in the United States 30 in Japan 10 in Germany and 20 in other countries

If cars are selected at random what is the probability that it is made in Japan or Germany

P(Japanese or German) = P(Japanese) + P(German) = 3 + 1 = 4

Suppose that 40 of cars in Fort Smith are manufactured in the United States 30 in Japan 10 in Germany and 20 in other countries

If cars are selected at random what is the probability that you see two in a row from Japan

P(2 Japanese in a row) = P(Japanese) and P(Japanese) = P(J) x P(J) = 3 x 3 = 09

Suppose that 40 of cars in Fort Smith are manufactured in the United States 30 in Japan 10 in Germany and 20 in other countries

If cars are selected at random what is the probability that none of three cars came from Germany

P(no Germany in three) = P(not G) x P(not G) x P(not G) = 9 x 9 x 9 = 729

Suppose that 40 of cars in Fort Smith are manufactured in the United States 30 in Japan 10 in Germany and 20 in other countries

If cars are selected at random what is the probability that at least one of three cars is US made

P(at least one US in three) = 1 ndash P(no US in three) = 1 ndash (6)(6)(6) = 784

Suppose that 40 of cars in Fort Smith are manufactured in the United States 30 in Japan 10 in Germany and 20 in other countries

If cars are selected at random what is the probability that the first Japanese car is the fourth one you choose

P(first J is the fourth car) = P(not J) x P(not J) x P(not J) x P(J) = (7)3 (3) = 1029

Watch out for

bull probabilities that donrsquot add up to 1

bull donrsquot add probabilities of events if they are not disjoint

bull donrsquot multiply probabilities of events if they are not independent

bull donrsquot confuse disjoint and independent

Rule 7 Conditional Probability

bull A probability that takes into account a given condition

P(A)

B)P(AA)|P(B

P(given)

P(and)A)|P(B

In a class there are 12 boys made up of 8 Seniors and 4 Juniors There are also 8 girls made up of 3 Seniors and 5 Juniors Find the probability of choosing a boy given that he is a Senior

Boy Girl Total

Senior 8 3 11

Junior 4 5 9

Total 12 8 20

P(Boy|Senior) = 811

P(Senior|Boy) = 812 = 23

Probabilities from two way tables

Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359

12) What is the probability that the driver is a student

359195

)( StudentP

Probabilities from two way tables

Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359

13) What is the probability that the driver drives a European car

35945

)( EuropeanP

Probabilities from two way tables

Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359

14) What is the probability that the driver drives an American or Asian car

Disjoint359102212

)(

AsianorAmericanP

Probabilities from two way tables

Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359

15) What is the probability that the driver is staff or drives an Asian car

Disjoint35947102164

)(

AsianorStaffP

Probabilities from two way tables

Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359

16) What is the probability that the driver is staff and drives an Asian car

35947

)( AsianandStaffP

Probabilities from two way tables

Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359

17) If the driver is a student what is the probability that they drive an American car

Condition195107

)|( StudentAmericanP

Probabilities from two way tables

Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359

18) What is the probability that the driver is a student if the driver drives a European car

Condition 4533

)|( EuropeanStudentP

Definition of Independent Events

Two events E and F are independent if and only if

P(F | E) = P(F) or P(E | F) = P(E)

EXAMPLE Illustrating Independent Events

The probability a randomly selected murder victim is male is 07515 The probability a randomly selected murder victim is male given that they are less than 18 years old is 06751

Since P(male) = 07515 and

P(male | lt 18 years old) = 06751

the events ldquomalerdquo and ldquoless than 18 years oldrdquo are not independent In fact knowing the victim is less than 18 years old decreases the probability that the victim is male

I draw one card and look

at it I tell you it is red What is the

probability it is a heart

P( heart | red) =

13P(heart and red) 152

26P(red) 252

Are ldquored cardrdquo and ldquospaderdquo mutually

exclusive Are they

independentA red card canrsquot be a spade so they ARE mutually exclusive

Are ldquored cardrdquo and

ldquoacerdquo mutually

exclusive Are they

independent2 aces are

red cards so they are

NOT mutually exclusive

Are ldquoface cardrdquo and

ldquokingrdquo mutually

exclusive Are they

independentKings are

Face cards so they are

NOT mutually exclusive

ldquoSlump I ainrsquot in no slump I just ainrsquot hittinrdquo

Yogi Berra

• PowerPoint Presentation
• Probability
• Law of Large Numbers
• Basic Rules of Probability
• Slide 5
• Slide 6
• Independent
• Slide 8
• Slide 9
• Slide 10
• Slide 11
• Slide 12
• Slide 13
• Slide 14
• Slide 15
• Slide 16
• Slide 17
• Slide 18
• Slide 19
• Slide 20
• Slide 21
• Slide 22
• Slide 23
• Slide 24
• Slide 25
• Slide 26
• Slide 27
• Slide 28
• Slide 29
• Slide 30
• Slide 31
• Slide 32
• Rule 7 Conditional Probability
• Slide 34
• Probabilities from two way tables
• Slide 36
• Slide 37
• Slide 38
• Slide 39
• Slide 40
• Slide 41
• Slide 42
• Slide 43
• Slide 44
• Slide 45
• Slide 46
• Slide 47
• Slide 48

Basic Rules of ProbabilityRule 1 Legitimate Values

For any event E 0 lt P(E) lt 1

A probabilityis a number between 0 and 1

The probability of rain must be 110

Rule 2 Sample spaceIf S is the sample space P(S) = 1

ldquoSomething Has to Happen Rulerdquo

The probability of the set of all possible outcomes must be 1

Irsquom 100 sure you are going to have a boyhellip or a girl

If the probability that you get to class on time is 8 then the probability that you do not get to class on time is 2

Rule 3 ComplementFor any event E

P(E) + P(not E) = 1 P(E) = 1 ndash P(not E)

Independent

bull Two events are independent if knowing that one will occur (or has occurred) does not change the probability that the other occursndash A randomly selected student is female - What is

the probability she plays soccer for SHS

ndash A randomly selected student is female - What is the probability she plays football for SHS

Independent

Dependent

Rule 4 Multiplication

If two events A amp B are independent

General rule

P(B) P(A) B) ampP(A

A)|P(B P(A) B) ampP(A If the probability of rolling a 5 on a fair dice is 16 what is the probability of rolling a 5 three times in a row

P( three 5rsquos in a row) = (16) x (16) x (16) = 1216 or 004629

)( BAP

Independent)( BAP

Yes

)()( BPAP

What does this mean

Given a deck of cards and a die

one card is drawn and the dice is rolled

What is the probability that an ace is drawn and an even is

rolledP(ace and even) = P(ace) P(even) =

4 3 1

52 6 26

)( BAP

Independent)( BAP

Yes No

)()( BPAP )|()( ABPAP

Given a deck of cards two

cards are drawn

without replacement What is the probability

that they are both heartsP(heart and heart) =

)|()( ABPAP

13 12 1

52 51 17

Ex 6) Suppose I will pick two cards from a standard deck without replacement What is the probability that I select two spades

Are the cards independent NO

P(A amp B) = P(A) P(B|A)

Read ldquoprobability of B given that A occursrdquo

P(Spade amp Spade) = 14 1251 = 117

The probability of getting a spade given that a spade has already been drawn

Rule 5 AdditionIf two events E amp F are disjoint

P(E or F) = P(E) + P(F)

If the probability that a randomly selected student is a junior (A) is 2 and the probability that the student is a senior (B) is 5 what is the probability that the student is either a junior or a senior

P(A υ B) = P(A) + P(B) if A and B are disjoint

P(A υ B) = 2 + 5 = 7

bull Two events that have no common outcomes are said to be disjoint or mutually exclusive

A and B are disjoint events

Rule 5 AdditionIf two events E amp F are disjoint P(E or F) = P(E) + P(F)

(General) If two events E amp F are not disjoint

P(E or F) = P(E) + P(F) ndash P(E amp F)Probability of owning a MP3 player 50Probability of owning a computer 90

So the probability of owning a MP3 player or a computer

is 140

Not disjoint events

)( FEP What does this mean

)( FEP Mutually exclusive

)()( FPEP

Yes

Given a deck of cards one

card is drawn What

is the probability

that it is a 3 or a 4

P(3 or 4) = P(3) + P(4) =

4 4 2

52 52 13

)( FEP

)( FEP Mutually exclusive

)()( FPEP

Yes No

)()()( FEPFPEP

Given a deck of cards one

card is drawn What

is the probability that it is an ace or a red

cardP(ace or red) = P(ace) + P(red) ndash P(ace and red) =

52

4

52

26

52

2

52

28+ - =

)( FEP

)( FEP Mutually exclusive

)()( FPEP

Yes No

)()()( FEPFPEP

Independent Yes )()( FPEP

Ex 5)

If P(A) = 045 P(B) = 035 and A amp B are independent find P(A or B)Is A amp B disjoint

If A amp B are disjoint are they independent

Disjoint events do not happen at the same time

So if A occurs can B occur

Disjoint events are dependent

NO independent events cannot be disjoint

P(A or B) = P(A) + P(B) ndash P(A amp B)

How can you find the

probability of A amp B

P(A or B) = 45 + 35 - 45(35) = 06425

If independent

multiply

In a class there are 12 boys made up of 8 Seniors and 4 Juniors There are also 8 girls made up of 3 Seniors and 5 Juniors Find the probability of choosing a boy or a Senior

Note that choosing a boy and choosing a Senior are not disjoint (they can occur simultaneously)

P(boy or a senior) = P(Boy) + P(Senior) ndash P(Senior boy) = 12 11 8 15 3

20 20 20 20 4

Boy Girl Total

Senior 8 3 11

Junior 4 5 9

Total 12 8 20

Rule 6 At least one

The probability that at least one outcome happens is 1 minus the probability that no outcomes happen

P(at least 1) = 1 ndash P(none)

For a sales promotion the manufacturer places winning symbols under the caps of 10 of all Dr Pepper bottles You buy a six-pack What is the probability that you win something

P(at least one winning symbol) =

1 ndash P(no winning symbols) 1 - 96 = 4686

Suppose that 40 of cars in Fort Smith are manufactured in the United States 30 in Japan 10 in Germany and 20 in other countries

If cars are selected at random what is the probability that it is not US made

P(not US made) = 1 ndash P(US made) = 1 - 4 = 6

Suppose that 40 of cars in Fort Smith are manufactured in the United States 30 in Japan 10 in Germany and 20 in other countries

If cars are selected at random what is the probability that it is made in Japan or Germany

P(Japanese or German) = P(Japanese) + P(German) = 3 + 1 = 4

Suppose that 40 of cars in Fort Smith are manufactured in the United States 30 in Japan 10 in Germany and 20 in other countries

If cars are selected at random what is the probability that you see two in a row from Japan

P(2 Japanese in a row) = P(Japanese) and P(Japanese) = P(J) x P(J) = 3 x 3 = 09

Suppose that 40 of cars in Fort Smith are manufactured in the United States 30 in Japan 10 in Germany and 20 in other countries

If cars are selected at random what is the probability that none of three cars came from Germany

P(no Germany in three) = P(not G) x P(not G) x P(not G) = 9 x 9 x 9 = 729

Suppose that 40 of cars in Fort Smith are manufactured in the United States 30 in Japan 10 in Germany and 20 in other countries

If cars are selected at random what is the probability that at least one of three cars is US made

P(at least one US in three) = 1 ndash P(no US in three) = 1 ndash (6)(6)(6) = 784

Suppose that 40 of cars in Fort Smith are manufactured in the United States 30 in Japan 10 in Germany and 20 in other countries

If cars are selected at random what is the probability that the first Japanese car is the fourth one you choose

P(first J is the fourth car) = P(not J) x P(not J) x P(not J) x P(J) = (7)3 (3) = 1029

Watch out for

bull probabilities that donrsquot add up to 1

bull donrsquot add probabilities of events if they are not disjoint

bull donrsquot multiply probabilities of events if they are not independent

bull donrsquot confuse disjoint and independent

Rule 7 Conditional Probability

bull A probability that takes into account a given condition

P(A)

B)P(AA)|P(B

P(given)

P(and)A)|P(B

In a class there are 12 boys made up of 8 Seniors and 4 Juniors There are also 8 girls made up of 3 Seniors and 5 Juniors Find the probability of choosing a boy given that he is a Senior

Boy Girl Total

Senior 8 3 11

Junior 4 5 9

Total 12 8 20

P(Boy|Senior) = 811

P(Senior|Boy) = 812 = 23

Probabilities from two way tables

Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359

12) What is the probability that the driver is a student

359195

)( StudentP

Probabilities from two way tables

Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359

13) What is the probability that the driver drives a European car

35945

)( EuropeanP

Probabilities from two way tables

Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359

14) What is the probability that the driver drives an American or Asian car

Disjoint359102212

)(

AsianorAmericanP

Probabilities from two way tables

Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359

15) What is the probability that the driver is staff or drives an Asian car

Disjoint35947102164

)(

AsianorStaffP

Probabilities from two way tables

Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359

16) What is the probability that the driver is staff and drives an Asian car

35947

)( AsianandStaffP

Probabilities from two way tables

Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359

17) If the driver is a student what is the probability that they drive an American car

Condition195107

)|( StudentAmericanP

Probabilities from two way tables

Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359

18) What is the probability that the driver is a student if the driver drives a European car

Condition 4533

)|( EuropeanStudentP

Definition of Independent Events

Two events E and F are independent if and only if

P(F | E) = P(F) or P(E | F) = P(E)

EXAMPLE Illustrating Independent Events

The probability a randomly selected murder victim is male is 07515 The probability a randomly selected murder victim is male given that they are less than 18 years old is 06751

Since P(male) = 07515 and

P(male | lt 18 years old) = 06751

the events ldquomalerdquo and ldquoless than 18 years oldrdquo are not independent In fact knowing the victim is less than 18 years old decreases the probability that the victim is male

I draw one card and look

at it I tell you it is red What is the

probability it is a heart

P( heart | red) =

13P(heart and red) 152

26P(red) 252

Are ldquored cardrdquo and ldquospaderdquo mutually

exclusive Are they

independentA red card canrsquot be a spade so they ARE mutually exclusive

Are ldquored cardrdquo and

ldquoacerdquo mutually

exclusive Are they

independent2 aces are

red cards so they are

NOT mutually exclusive

Are ldquoface cardrdquo and

ldquokingrdquo mutually

exclusive Are they

independentKings are

Face cards so they are

NOT mutually exclusive

ldquoSlump I ainrsquot in no slump I just ainrsquot hittinrdquo

Yogi Berra

• PowerPoint Presentation
• Probability
• Law of Large Numbers
• Basic Rules of Probability
• Slide 5
• Slide 6
• Independent
• Slide 8
• Slide 9
• Slide 10
• Slide 11
• Slide 12
• Slide 13
• Slide 14
• Slide 15
• Slide 16
• Slide 17
• Slide 18
• Slide 19
• Slide 20
• Slide 21
• Slide 22
• Slide 23
• Slide 24
• Slide 25
• Slide 26
• Slide 27
• Slide 28
• Slide 29
• Slide 30
• Slide 31
• Slide 32
• Rule 7 Conditional Probability
• Slide 34
• Probabilities from two way tables
• Slide 36
• Slide 37
• Slide 38
• Slide 39
• Slide 40
• Slide 41
• Slide 42
• Slide 43
• Slide 44
• Slide 45
• Slide 46
• Slide 47
• Slide 48

Rule 2 Sample spaceIf S is the sample space P(S) = 1

ldquoSomething Has to Happen Rulerdquo

The probability of the set of all possible outcomes must be 1

Irsquom 100 sure you are going to have a boyhellip or a girl

If the probability that you get to class on time is 8 then the probability that you do not get to class on time is 2

Rule 3 ComplementFor any event E

P(E) + P(not E) = 1 P(E) = 1 ndash P(not E)

Independent

bull Two events are independent if knowing that one will occur (or has occurred) does not change the probability that the other occursndash A randomly selected student is female - What is

the probability she plays soccer for SHS

ndash A randomly selected student is female - What is the probability she plays football for SHS

Independent

Dependent

Rule 4 Multiplication

If two events A amp B are independent

General rule

P(B) P(A) B) ampP(A

A)|P(B P(A) B) ampP(A If the probability of rolling a 5 on a fair dice is 16 what is the probability of rolling a 5 three times in a row

P( three 5rsquos in a row) = (16) x (16) x (16) = 1216 or 004629

)( BAP

Independent)( BAP

Yes

)()( BPAP

What does this mean

Given a deck of cards and a die

one card is drawn and the dice is rolled

What is the probability that an ace is drawn and an even is

rolledP(ace and even) = P(ace) P(even) =

4 3 1

52 6 26

)( BAP

Independent)( BAP

Yes No

)()( BPAP )|()( ABPAP

Given a deck of cards two

cards are drawn

without replacement What is the probability

that they are both heartsP(heart and heart) =

)|()( ABPAP

13 12 1

52 51 17

Ex 6) Suppose I will pick two cards from a standard deck without replacement What is the probability that I select two spades

Are the cards independent NO

P(A amp B) = P(A) P(B|A)

Read ldquoprobability of B given that A occursrdquo

P(Spade amp Spade) = 14 1251 = 117

The probability of getting a spade given that a spade has already been drawn

Rule 5 AdditionIf two events E amp F are disjoint

P(E or F) = P(E) + P(F)

If the probability that a randomly selected student is a junior (A) is 2 and the probability that the student is a senior (B) is 5 what is the probability that the student is either a junior or a senior

P(A υ B) = P(A) + P(B) if A and B are disjoint

P(A υ B) = 2 + 5 = 7

bull Two events that have no common outcomes are said to be disjoint or mutually exclusive

A and B are disjoint events

Rule 5 AdditionIf two events E amp F are disjoint P(E or F) = P(E) + P(F)

(General) If two events E amp F are not disjoint

P(E or F) = P(E) + P(F) ndash P(E amp F)Probability of owning a MP3 player 50Probability of owning a computer 90

So the probability of owning a MP3 player or a computer

is 140

Not disjoint events

)( FEP What does this mean

)( FEP Mutually exclusive

)()( FPEP

Yes

Given a deck of cards one

card is drawn What

is the probability

that it is a 3 or a 4

P(3 or 4) = P(3) + P(4) =

4 4 2

52 52 13

)( FEP

)( FEP Mutually exclusive

)()( FPEP

Yes No

)()()( FEPFPEP

Given a deck of cards one

card is drawn What

is the probability that it is an ace or a red

cardP(ace or red) = P(ace) + P(red) ndash P(ace and red) =

52

4

52

26

52

2

52

28+ - =

)( FEP

)( FEP Mutually exclusive

)()( FPEP

Yes No

)()()( FEPFPEP

Independent Yes )()( FPEP

Ex 5)

If P(A) = 045 P(B) = 035 and A amp B are independent find P(A or B)Is A amp B disjoint

If A amp B are disjoint are they independent

Disjoint events do not happen at the same time

So if A occurs can B occur

Disjoint events are dependent

NO independent events cannot be disjoint

P(A or B) = P(A) + P(B) ndash P(A amp B)

How can you find the

probability of A amp B

P(A or B) = 45 + 35 - 45(35) = 06425

If independent

multiply

In a class there are 12 boys made up of 8 Seniors and 4 Juniors There are also 8 girls made up of 3 Seniors and 5 Juniors Find the probability of choosing a boy or a Senior

Note that choosing a boy and choosing a Senior are not disjoint (they can occur simultaneously)

P(boy or a senior) = P(Boy) + P(Senior) ndash P(Senior boy) = 12 11 8 15 3

20 20 20 20 4

Boy Girl Total

Senior 8 3 11

Junior 4 5 9

Total 12 8 20

Rule 6 At least one

The probability that at least one outcome happens is 1 minus the probability that no outcomes happen

P(at least 1) = 1 ndash P(none)

For a sales promotion the manufacturer places winning symbols under the caps of 10 of all Dr Pepper bottles You buy a six-pack What is the probability that you win something

P(at least one winning symbol) =

1 ndash P(no winning symbols) 1 - 96 = 4686

Suppose that 40 of cars in Fort Smith are manufactured in the United States 30 in Japan 10 in Germany and 20 in other countries

If cars are selected at random what is the probability that it is not US made

P(not US made) = 1 ndash P(US made) = 1 - 4 = 6

Suppose that 40 of cars in Fort Smith are manufactured in the United States 30 in Japan 10 in Germany and 20 in other countries

If cars are selected at random what is the probability that it is made in Japan or Germany

P(Japanese or German) = P(Japanese) + P(German) = 3 + 1 = 4

Suppose that 40 of cars in Fort Smith are manufactured in the United States 30 in Japan 10 in Germany and 20 in other countries

If cars are selected at random what is the probability that you see two in a row from Japan

P(2 Japanese in a row) = P(Japanese) and P(Japanese) = P(J) x P(J) = 3 x 3 = 09

Suppose that 40 of cars in Fort Smith are manufactured in the United States 30 in Japan 10 in Germany and 20 in other countries

If cars are selected at random what is the probability that none of three cars came from Germany

P(no Germany in three) = P(not G) x P(not G) x P(not G) = 9 x 9 x 9 = 729

Suppose that 40 of cars in Fort Smith are manufactured in the United States 30 in Japan 10 in Germany and 20 in other countries

If cars are selected at random what is the probability that at least one of three cars is US made

P(at least one US in three) = 1 ndash P(no US in three) = 1 ndash (6)(6)(6) = 784

Suppose that 40 of cars in Fort Smith are manufactured in the United States 30 in Japan 10 in Germany and 20 in other countries

If cars are selected at random what is the probability that the first Japanese car is the fourth one you choose

P(first J is the fourth car) = P(not J) x P(not J) x P(not J) x P(J) = (7)3 (3) = 1029

Watch out for

bull probabilities that donrsquot add up to 1

bull donrsquot add probabilities of events if they are not disjoint

bull donrsquot multiply probabilities of events if they are not independent

bull donrsquot confuse disjoint and independent

Rule 7 Conditional Probability

bull A probability that takes into account a given condition

P(A)

B)P(AA)|P(B

P(given)

P(and)A)|P(B

In a class there are 12 boys made up of 8 Seniors and 4 Juniors There are also 8 girls made up of 3 Seniors and 5 Juniors Find the probability of choosing a boy given that he is a Senior

Boy Girl Total

Senior 8 3 11

Junior 4 5 9

Total 12 8 20

P(Boy|Senior) = 811

P(Senior|Boy) = 812 = 23

Probabilities from two way tables

Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359

12) What is the probability that the driver is a student

359195

)( StudentP

Probabilities from two way tables

Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359

13) What is the probability that the driver drives a European car

35945

)( EuropeanP

Probabilities from two way tables

Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359

14) What is the probability that the driver drives an American or Asian car

Disjoint359102212

)(

AsianorAmericanP

Probabilities from two way tables

Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359

15) What is the probability that the driver is staff or drives an Asian car

Disjoint35947102164

)(

AsianorStaffP

Probabilities from two way tables

Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359

16) What is the probability that the driver is staff and drives an Asian car

35947

)( AsianandStaffP

Probabilities from two way tables

Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359

17) If the driver is a student what is the probability that they drive an American car

Condition195107

)|( StudentAmericanP

Probabilities from two way tables

Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359

18) What is the probability that the driver is a student if the driver drives a European car

Condition 4533

)|( EuropeanStudentP

Definition of Independent Events

Two events E and F are independent if and only if

P(F | E) = P(F) or P(E | F) = P(E)

EXAMPLE Illustrating Independent Events

The probability a randomly selected murder victim is male is 07515 The probability a randomly selected murder victim is male given that they are less than 18 years old is 06751

Since P(male) = 07515 and

P(male | lt 18 years old) = 06751

the events ldquomalerdquo and ldquoless than 18 years oldrdquo are not independent In fact knowing the victim is less than 18 years old decreases the probability that the victim is male

I draw one card and look

at it I tell you it is red What is the

probability it is a heart

P( heart | red) =

13P(heart and red) 152

26P(red) 252

Are ldquored cardrdquo and ldquospaderdquo mutually

exclusive Are they

independentA red card canrsquot be a spade so they ARE mutually exclusive

Are ldquored cardrdquo and

ldquoacerdquo mutually

exclusive Are they

independent2 aces are

red cards so they are

NOT mutually exclusive

Are ldquoface cardrdquo and

ldquokingrdquo mutually

exclusive Are they

independentKings are

Face cards so they are

NOT mutually exclusive

ldquoSlump I ainrsquot in no slump I just ainrsquot hittinrdquo

Yogi Berra

• PowerPoint Presentation
• Probability
• Law of Large Numbers
• Basic Rules of Probability
• Slide 5
• Slide 6
• Independent
• Slide 8
• Slide 9
• Slide 10
• Slide 11
• Slide 12
• Slide 13
• Slide 14
• Slide 15
• Slide 16
• Slide 17
• Slide 18
• Slide 19
• Slide 20
• Slide 21
• Slide 22
• Slide 23
• Slide 24
• Slide 25
• Slide 26
• Slide 27
• Slide 28
• Slide 29
• Slide 30
• Slide 31
• Slide 32
• Rule 7 Conditional Probability
• Slide 34
• Probabilities from two way tables
• Slide 36
• Slide 37
• Slide 38
• Slide 39
• Slide 40
• Slide 41
• Slide 42
• Slide 43
• Slide 44
• Slide 45
• Slide 46
• Slide 47
• Slide 48

If the probability that you get to class on time is 8 then the probability that you do not get to class on time is 2

Rule 3 ComplementFor any event E

P(E) + P(not E) = 1 P(E) = 1 ndash P(not E)

Independent

bull Two events are independent if knowing that one will occur (or has occurred) does not change the probability that the other occursndash A randomly selected student is female - What is

the probability she plays soccer for SHS

ndash A randomly selected student is female - What is the probability she plays football for SHS

Independent

Dependent

Rule 4 Multiplication

If two events A amp B are independent

General rule

P(B) P(A) B) ampP(A

A)|P(B P(A) B) ampP(A If the probability of rolling a 5 on a fair dice is 16 what is the probability of rolling a 5 three times in a row

P( three 5rsquos in a row) = (16) x (16) x (16) = 1216 or 004629

)( BAP

Independent)( BAP

Yes

)()( BPAP

What does this mean

Given a deck of cards and a die

one card is drawn and the dice is rolled

What is the probability that an ace is drawn and an even is

rolledP(ace and even) = P(ace) P(even) =

4 3 1

52 6 26

)( BAP

Independent)( BAP

Yes No

)()( BPAP )|()( ABPAP

Given a deck of cards two

cards are drawn

without replacement What is the probability

that they are both heartsP(heart and heart) =

)|()( ABPAP

13 12 1

52 51 17

Ex 6) Suppose I will pick two cards from a standard deck without replacement What is the probability that I select two spades

Are the cards independent NO

P(A amp B) = P(A) P(B|A)

Read ldquoprobability of B given that A occursrdquo

P(Spade amp Spade) = 14 1251 = 117

The probability of getting a spade given that a spade has already been drawn

Rule 5 AdditionIf two events E amp F are disjoint

P(E or F) = P(E) + P(F)

If the probability that a randomly selected student is a junior (A) is 2 and the probability that the student is a senior (B) is 5 what is the probability that the student is either a junior or a senior

P(A υ B) = P(A) + P(B) if A and B are disjoint

P(A υ B) = 2 + 5 = 7

bull Two events that have no common outcomes are said to be disjoint or mutually exclusive

A and B are disjoint events

Rule 5 AdditionIf two events E amp F are disjoint P(E or F) = P(E) + P(F)

(General) If two events E amp F are not disjoint

P(E or F) = P(E) + P(F) ndash P(E amp F)Probability of owning a MP3 player 50Probability of owning a computer 90

So the probability of owning a MP3 player or a computer

is 140

Not disjoint events

)( FEP What does this mean

)( FEP Mutually exclusive

)()( FPEP

Yes

Given a deck of cards one

card is drawn What

is the probability

that it is a 3 or a 4

P(3 or 4) = P(3) + P(4) =

4 4 2

52 52 13

)( FEP

)( FEP Mutually exclusive

)()( FPEP

Yes No

)()()( FEPFPEP

Given a deck of cards one

card is drawn What

is the probability that it is an ace or a red

cardP(ace or red) = P(ace) + P(red) ndash P(ace and red) =

52

4

52

26

52

2

52

28+ - =

)( FEP

)( FEP Mutually exclusive

)()( FPEP

Yes No

)()()( FEPFPEP

Independent Yes )()( FPEP

Ex 5)

If P(A) = 045 P(B) = 035 and A amp B are independent find P(A or B)Is A amp B disjoint

If A amp B are disjoint are they independent

Disjoint events do not happen at the same time

So if A occurs can B occur

Disjoint events are dependent

NO independent events cannot be disjoint

P(A or B) = P(A) + P(B) ndash P(A amp B)

How can you find the

probability of A amp B

P(A or B) = 45 + 35 - 45(35) = 06425

If independent

multiply

In a class there are 12 boys made up of 8 Seniors and 4 Juniors There are also 8 girls made up of 3 Seniors and 5 Juniors Find the probability of choosing a boy or a Senior

Note that choosing a boy and choosing a Senior are not disjoint (they can occur simultaneously)

P(boy or a senior) = P(Boy) + P(Senior) ndash P(Senior boy) = 12 11 8 15 3

20 20 20 20 4

Boy Girl Total

Senior 8 3 11

Junior 4 5 9

Total 12 8 20

Rule 6 At least one

The probability that at least one outcome happens is 1 minus the probability that no outcomes happen

P(at least 1) = 1 ndash P(none)

For a sales promotion the manufacturer places winning symbols under the caps of 10 of all Dr Pepper bottles You buy a six-pack What is the probability that you win something

P(at least one winning symbol) =

1 ndash P(no winning symbols) 1 - 96 = 4686

Suppose that 40 of cars in Fort Smith are manufactured in the United States 30 in Japan 10 in Germany and 20 in other countries

If cars are selected at random what is the probability that it is not US made

P(not US made) = 1 ndash P(US made) = 1 - 4 = 6

Suppose that 40 of cars in Fort Smith are manufactured in the United States 30 in Japan 10 in Germany and 20 in other countries

If cars are selected at random what is the probability that it is made in Japan or Germany

P(Japanese or German) = P(Japanese) + P(German) = 3 + 1 = 4

Suppose that 40 of cars in Fort Smith are manufactured in the United States 30 in Japan 10 in Germany and 20 in other countries

If cars are selected at random what is the probability that you see two in a row from Japan

P(2 Japanese in a row) = P(Japanese) and P(Japanese) = P(J) x P(J) = 3 x 3 = 09

Suppose that 40 of cars in Fort Smith are manufactured in the United States 30 in Japan 10 in Germany and 20 in other countries

If cars are selected at random what is the probability that none of three cars came from Germany

P(no Germany in three) = P(not G) x P(not G) x P(not G) = 9 x 9 x 9 = 729

Suppose that 40 of cars in Fort Smith are manufactured in the United States 30 in Japan 10 in Germany and 20 in other countries

If cars are selected at random what is the probability that at least one of three cars is US made

P(at least one US in three) = 1 ndash P(no US in three) = 1 ndash (6)(6)(6) = 784

Suppose that 40 of cars in Fort Smith are manufactured in the United States 30 in Japan 10 in Germany and 20 in other countries

If cars are selected at random what is the probability that the first Japanese car is the fourth one you choose

P(first J is the fourth car) = P(not J) x P(not J) x P(not J) x P(J) = (7)3 (3) = 1029

Watch out for

bull probabilities that donrsquot add up to 1

bull donrsquot add probabilities of events if they are not disjoint

bull donrsquot multiply probabilities of events if they are not independent

bull donrsquot confuse disjoint and independent

Rule 7 Conditional Probability

bull A probability that takes into account a given condition

P(A)

B)P(AA)|P(B

P(given)

P(and)A)|P(B

In a class there are 12 boys made up of 8 Seniors and 4 Juniors There are also 8 girls made up of 3 Seniors and 5 Juniors Find the probability of choosing a boy given that he is a Senior

Boy Girl Total

Senior 8 3 11

Junior 4 5 9

Total 12 8 20

P(Boy|Senior) = 811

P(Senior|Boy) = 812 = 23

Probabilities from two way tables

Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359

12) What is the probability that the driver is a student

359195

)( StudentP

Probabilities from two way tables

Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359

13) What is the probability that the driver drives a European car

35945

)( EuropeanP

Probabilities from two way tables

Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359

14) What is the probability that the driver drives an American or Asian car

Disjoint359102212

)(

AsianorAmericanP

Probabilities from two way tables

Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359

15) What is the probability that the driver is staff or drives an Asian car

Disjoint35947102164

)(

AsianorStaffP

Probabilities from two way tables

Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359

16) What is the probability that the driver is staff and drives an Asian car

35947

)( AsianandStaffP

Probabilities from two way tables

Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359

17) If the driver is a student what is the probability that they drive an American car

Condition195107

)|( StudentAmericanP

Probabilities from two way tables

Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359

18) What is the probability that the driver is a student if the driver drives a European car

Condition 4533

)|( EuropeanStudentP

Definition of Independent Events

Two events E and F are independent if and only if

P(F | E) = P(F) or P(E | F) = P(E)

EXAMPLE Illustrating Independent Events

The probability a randomly selected murder victim is male is 07515 The probability a randomly selected murder victim is male given that they are less than 18 years old is 06751

Since P(male) = 07515 and

P(male | lt 18 years old) = 06751

the events ldquomalerdquo and ldquoless than 18 years oldrdquo are not independent In fact knowing the victim is less than 18 years old decreases the probability that the victim is male

I draw one card and look

at it I tell you it is red What is the

probability it is a heart

P( heart | red) =

13P(heart and red) 152

26P(red) 252

Are ldquored cardrdquo and ldquospaderdquo mutually

exclusive Are they

independentA red card canrsquot be a spade so they ARE mutually exclusive

Are ldquored cardrdquo and

ldquoacerdquo mutually

exclusive Are they

independent2 aces are

red cards so they are

NOT mutually exclusive

Are ldquoface cardrdquo and

ldquokingrdquo mutually

exclusive Are they

independentKings are

Face cards so they are

NOT mutually exclusive

ldquoSlump I ainrsquot in no slump I just ainrsquot hittinrdquo

Yogi Berra

• PowerPoint Presentation
• Probability
• Law of Large Numbers
• Basic Rules of Probability
• Slide 5
• Slide 6
• Independent
• Slide 8
• Slide 9
• Slide 10
• Slide 11
• Slide 12
• Slide 13
• Slide 14
• Slide 15
• Slide 16
• Slide 17
• Slide 18
• Slide 19
• Slide 20
• Slide 21
• Slide 22
• Slide 23
• Slide 24
• Slide 25
• Slide 26
• Slide 27
• Slide 28
• Slide 29
• Slide 30
• Slide 31
• Slide 32
• Rule 7 Conditional Probability
• Slide 34
• Probabilities from two way tables
• Slide 36
• Slide 37
• Slide 38
• Slide 39
• Slide 40
• Slide 41
• Slide 42
• Slide 43
• Slide 44
• Slide 45
• Slide 46
• Slide 47
• Slide 48

Independent

bull Two events are independent if knowing that one will occur (or has occurred) does not change the probability that the other occursndash A randomly selected student is female - What is

the probability she plays soccer for SHS

ndash A randomly selected student is female - What is the probability she plays football for SHS

Independent

Dependent

Rule 4 Multiplication

If two events A amp B are independent

General rule

P(B) P(A) B) ampP(A

A)|P(B P(A) B) ampP(A If the probability of rolling a 5 on a fair dice is 16 what is the probability of rolling a 5 three times in a row

P( three 5rsquos in a row) = (16) x (16) x (16) = 1216 or 004629

)( BAP

Independent)( BAP

Yes

)()( BPAP

What does this mean

Given a deck of cards and a die

one card is drawn and the dice is rolled

What is the probability that an ace is drawn and an even is

rolledP(ace and even) = P(ace) P(even) =

4 3 1

52 6 26

)( BAP

Independent)( BAP

Yes No

)()( BPAP )|()( ABPAP

Given a deck of cards two

cards are drawn

without replacement What is the probability

that they are both heartsP(heart and heart) =

)|()( ABPAP

13 12 1

52 51 17

Ex 6) Suppose I will pick two cards from a standard deck without replacement What is the probability that I select two spades

Are the cards independent NO

P(A amp B) = P(A) P(B|A)

Read ldquoprobability of B given that A occursrdquo

P(Spade amp Spade) = 14 1251 = 117

The probability of getting a spade given that a spade has already been drawn

Rule 5 AdditionIf two events E amp F are disjoint

P(E or F) = P(E) + P(F)

If the probability that a randomly selected student is a junior (A) is 2 and the probability that the student is a senior (B) is 5 what is the probability that the student is either a junior or a senior

P(A υ B) = P(A) + P(B) if A and B are disjoint

P(A υ B) = 2 + 5 = 7

bull Two events that have no common outcomes are said to be disjoint or mutually exclusive

A and B are disjoint events

Rule 5 AdditionIf two events E amp F are disjoint P(E or F) = P(E) + P(F)

(General) If two events E amp F are not disjoint

P(E or F) = P(E) + P(F) ndash P(E amp F)Probability of owning a MP3 player 50Probability of owning a computer 90

So the probability of owning a MP3 player or a computer

is 140

Not disjoint events

)( FEP What does this mean

)( FEP Mutually exclusive

)()( FPEP

Yes

Given a deck of cards one

card is drawn What

is the probability

that it is a 3 or a 4

P(3 or 4) = P(3) + P(4) =

4 4 2

52 52 13

)( FEP

)( FEP Mutually exclusive

)()( FPEP

Yes No

)()()( FEPFPEP

Given a deck of cards one

card is drawn What

is the probability that it is an ace or a red

cardP(ace or red) = P(ace) + P(red) ndash P(ace and red) =

52

4

52

26

52

2

52

28+ - =

)( FEP

)( FEP Mutually exclusive

)()( FPEP

Yes No

)()()( FEPFPEP

Independent Yes )()( FPEP

Ex 5)

If P(A) = 045 P(B) = 035 and A amp B are independent find P(A or B)Is A amp B disjoint

If A amp B are disjoint are they independent

Disjoint events do not happen at the same time

So if A occurs can B occur

Disjoint events are dependent

NO independent events cannot be disjoint

P(A or B) = P(A) + P(B) ndash P(A amp B)

How can you find the

probability of A amp B

P(A or B) = 45 + 35 - 45(35) = 06425

If independent

multiply

In a class there are 12 boys made up of 8 Seniors and 4 Juniors There are also 8 girls made up of 3 Seniors and 5 Juniors Find the probability of choosing a boy or a Senior

Note that choosing a boy and choosing a Senior are not disjoint (they can occur simultaneously)

P(boy or a senior) = P(Boy) + P(Senior) ndash P(Senior boy) = 12 11 8 15 3

20 20 20 20 4

Boy Girl Total

Senior 8 3 11

Junior 4 5 9

Total 12 8 20

Rule 6 At least one

The probability that at least one outcome happens is 1 minus the probability that no outcomes happen

P(at least 1) = 1 ndash P(none)

For a sales promotion the manufacturer places winning symbols under the caps of 10 of all Dr Pepper bottles You buy a six-pack What is the probability that you win something

P(at least one winning symbol) =

1 ndash P(no winning symbols) 1 - 96 = 4686

Suppose that 40 of cars in Fort Smith are manufactured in the United States 30 in Japan 10 in Germany and 20 in other countries

If cars are selected at random what is the probability that it is not US made

P(not US made) = 1 ndash P(US made) = 1 - 4 = 6

Suppose that 40 of cars in Fort Smith are manufactured in the United States 30 in Japan 10 in Germany and 20 in other countries

If cars are selected at random what is the probability that it is made in Japan or Germany

P(Japanese or German) = P(Japanese) + P(German) = 3 + 1 = 4

Suppose that 40 of cars in Fort Smith are manufactured in the United States 30 in Japan 10 in Germany and 20 in other countries

If cars are selected at random what is the probability that you see two in a row from Japan

P(2 Japanese in a row) = P(Japanese) and P(Japanese) = P(J) x P(J) = 3 x 3 = 09

Suppose that 40 of cars in Fort Smith are manufactured in the United States 30 in Japan 10 in Germany and 20 in other countries

If cars are selected at random what is the probability that none of three cars came from Germany

P(no Germany in three) = P(not G) x P(not G) x P(not G) = 9 x 9 x 9 = 729

Suppose that 40 of cars in Fort Smith are manufactured in the United States 30 in Japan 10 in Germany and 20 in other countries

If cars are selected at random what is the probability that at least one of three cars is US made

P(at least one US in three) = 1 ndash P(no US in three) = 1 ndash (6)(6)(6) = 784

Suppose that 40 of cars in Fort Smith are manufactured in the United States 30 in Japan 10 in Germany and 20 in other countries

If cars are selected at random what is the probability that the first Japanese car is the fourth one you choose

P(first J is the fourth car) = P(not J) x P(not J) x P(not J) x P(J) = (7)3 (3) = 1029

Watch out for

bull probabilities that donrsquot add up to 1

bull donrsquot add probabilities of events if they are not disjoint

bull donrsquot multiply probabilities of events if they are not independent

bull donrsquot confuse disjoint and independent

Rule 7 Conditional Probability

bull A probability that takes into account a given condition

P(A)

B)P(AA)|P(B

P(given)

P(and)A)|P(B

In a class there are 12 boys made up of 8 Seniors and 4 Juniors There are also 8 girls made up of 3 Seniors and 5 Juniors Find the probability of choosing a boy given that he is a Senior

Boy Girl Total

Senior 8 3 11

Junior 4 5 9

Total 12 8 20

P(Boy|Senior) = 811

P(Senior|Boy) = 812 = 23

Probabilities from two way tables

Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359

12) What is the probability that the driver is a student

359195

)( StudentP

Probabilities from two way tables

Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359

13) What is the probability that the driver drives a European car

35945

)( EuropeanP

Probabilities from two way tables

Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359

14) What is the probability that the driver drives an American or Asian car

Disjoint359102212

)(

AsianorAmericanP

Probabilities from two way tables

Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359

15) What is the probability that the driver is staff or drives an Asian car

Disjoint35947102164

)(

AsianorStaffP

Probabilities from two way tables

Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359

16) What is the probability that the driver is staff and drives an Asian car

35947

)( AsianandStaffP

Probabilities from two way tables

Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359

17) If the driver is a student what is the probability that they drive an American car

Condition195107

)|( StudentAmericanP

Probabilities from two way tables

Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359

18) What is the probability that the driver is a student if the driver drives a European car

Condition 4533

)|( EuropeanStudentP

Definition of Independent Events

Two events E and F are independent if and only if

P(F | E) = P(F) or P(E | F) = P(E)

EXAMPLE Illustrating Independent Events

The probability a randomly selected murder victim is male is 07515 The probability a randomly selected murder victim is male given that they are less than 18 years old is 06751

Since P(male) = 07515 and

P(male | lt 18 years old) = 06751

the events ldquomalerdquo and ldquoless than 18 years oldrdquo are not independent In fact knowing the victim is less than 18 years old decreases the probability that the victim is male

I draw one card and look

at it I tell you it is red What is the

probability it is a heart

P( heart | red) =

13P(heart and red) 152

26P(red) 252

Are ldquored cardrdquo and ldquospaderdquo mutually

exclusive Are they

independentA red card canrsquot be a spade so they ARE mutually exclusive

Are ldquored cardrdquo and

ldquoacerdquo mutually

exclusive Are they

independent2 aces are

red cards so they are

NOT mutually exclusive

Are ldquoface cardrdquo and

ldquokingrdquo mutually

exclusive Are they

independentKings are

Face cards so they are

NOT mutually exclusive

ldquoSlump I ainrsquot in no slump I just ainrsquot hittinrdquo

Yogi Berra

• PowerPoint Presentation
• Probability
• Law of Large Numbers
• Basic Rules of Probability
• Slide 5
• Slide 6
• Independent
• Slide 8
• Slide 9
• Slide 10
• Slide 11
• Slide 12
• Slide 13
• Slide 14
• Slide 15
• Slide 16
• Slide 17
• Slide 18
• Slide 19
• Slide 20
• Slide 21
• Slide 22
• Slide 23
• Slide 24
• Slide 25
• Slide 26
• Slide 27
• Slide 28
• Slide 29
• Slide 30
• Slide 31
• Slide 32
• Rule 7 Conditional Probability
• Slide 34
• Probabilities from two way tables
• Slide 36
• Slide 37
• Slide 38
• Slide 39
• Slide 40
• Slide 41
• Slide 42
• Slide 43
• Slide 44
• Slide 45
• Slide 46
• Slide 47
• Slide 48

Rule 4 Multiplication

If two events A amp B are independent

General rule

P(B) P(A) B) ampP(A

A)|P(B P(A) B) ampP(A If the probability of rolling a 5 on a fair dice is 16 what is the probability of rolling a 5 three times in a row

P( three 5rsquos in a row) = (16) x (16) x (16) = 1216 or 004629

)( BAP

Independent)( BAP

Yes

)()( BPAP

What does this mean

Given a deck of cards and a die

one card is drawn and the dice is rolled

What is the probability that an ace is drawn and an even is

rolledP(ace and even) = P(ace) P(even) =

4 3 1

52 6 26

)( BAP

Independent)( BAP

Yes No

)()( BPAP )|()( ABPAP

Given a deck of cards two

cards are drawn

without replacement What is the probability

that they are both heartsP(heart and heart) =

)|()( ABPAP

13 12 1

52 51 17

Ex 6) Suppose I will pick two cards from a standard deck without replacement What is the probability that I select two spades

Are the cards independent NO

P(A amp B) = P(A) P(B|A)

Read ldquoprobability of B given that A occursrdquo

P(Spade amp Spade) = 14 1251 = 117

The probability of getting a spade given that a spade has already been drawn

Rule 5 AdditionIf two events E amp F are disjoint

P(E or F) = P(E) + P(F)

If the probability that a randomly selected student is a junior (A) is 2 and the probability that the student is a senior (B) is 5 what is the probability that the student is either a junior or a senior

P(A υ B) = P(A) + P(B) if A and B are disjoint

P(A υ B) = 2 + 5 = 7

bull Two events that have no common outcomes are said to be disjoint or mutually exclusive

A and B are disjoint events

Rule 5 AdditionIf two events E amp F are disjoint P(E or F) = P(E) + P(F)

(General) If two events E amp F are not disjoint

P(E or F) = P(E) + P(F) ndash P(E amp F)Probability of owning a MP3 player 50Probability of owning a computer 90

So the probability of owning a MP3 player or a computer

is 140

Not disjoint events

)( FEP What does this mean

)( FEP Mutually exclusive

)()( FPEP

Yes

Given a deck of cards one

card is drawn What

is the probability

that it is a 3 or a 4

P(3 or 4) = P(3) + P(4) =

4 4 2

52 52 13

)( FEP

)( FEP Mutually exclusive

)()( FPEP

Yes No

)()()( FEPFPEP

Given a deck of cards one

card is drawn What

is the probability that it is an ace or a red

cardP(ace or red) = P(ace) + P(red) ndash P(ace and red) =

52

4

52

26

52

2

52

28+ - =

)( FEP

)( FEP Mutually exclusive

)()( FPEP

Yes No

)()()( FEPFPEP

Independent Yes )()( FPEP

Ex 5)

If P(A) = 045 P(B) = 035 and A amp B are independent find P(A or B)Is A amp B disjoint

If A amp B are disjoint are they independent

Disjoint events do not happen at the same time

So if A occurs can B occur

Disjoint events are dependent

NO independent events cannot be disjoint

P(A or B) = P(A) + P(B) ndash P(A amp B)

How can you find the

probability of A amp B

P(A or B) = 45 + 35 - 45(35) = 06425

If independent

multiply

In a class there are 12 boys made up of 8 Seniors and 4 Juniors There are also 8 girls made up of 3 Seniors and 5 Juniors Find the probability of choosing a boy or a Senior

Note that choosing a boy and choosing a Senior are not disjoint (they can occur simultaneously)

P(boy or a senior) = P(Boy) + P(Senior) ndash P(Senior boy) = 12 11 8 15 3

20 20 20 20 4

Boy Girl Total

Senior 8 3 11

Junior 4 5 9

Total 12 8 20

Rule 6 At least one

The probability that at least one outcome happens is 1 minus the probability that no outcomes happen

P(at least 1) = 1 ndash P(none)

For a sales promotion the manufacturer places winning symbols under the caps of 10 of all Dr Pepper bottles You buy a six-pack What is the probability that you win something

P(at least one winning symbol) =

1 ndash P(no winning symbols) 1 - 96 = 4686

Suppose that 40 of cars in Fort Smith are manufactured in the United States 30 in Japan 10 in Germany and 20 in other countries

If cars are selected at random what is the probability that it is not US made

P(not US made) = 1 ndash P(US made) = 1 - 4 = 6

Suppose that 40 of cars in Fort Smith are manufactured in the United States 30 in Japan 10 in Germany and 20 in other countries

If cars are selected at random what is the probability that it is made in Japan or Germany

P(Japanese or German) = P(Japanese) + P(German) = 3 + 1 = 4

Suppose that 40 of cars in Fort Smith are manufactured in the United States 30 in Japan 10 in Germany and 20 in other countries

If cars are selected at random what is the probability that you see two in a row from Japan

P(2 Japanese in a row) = P(Japanese) and P(Japanese) = P(J) x P(J) = 3 x 3 = 09

Suppose that 40 of cars in Fort Smith are manufactured in the United States 30 in Japan 10 in Germany and 20 in other countries

If cars are selected at random what is the probability that none of three cars came from Germany

P(no Germany in three) = P(not G) x P(not G) x P(not G) = 9 x 9 x 9 = 729

Suppose that 40 of cars in Fort Smith are manufactured in the United States 30 in Japan 10 in Germany and 20 in other countries

If cars are selected at random what is the probability that at least one of three cars is US made

P(at least one US in three) = 1 ndash P(no US in three) = 1 ndash (6)(6)(6) = 784

Suppose that 40 of cars in Fort Smith are manufactured in the United States 30 in Japan 10 in Germany and 20 in other countries

If cars are selected at random what is the probability that the first Japanese car is the fourth one you choose

P(first J is the fourth car) = P(not J) x P(not J) x P(not J) x P(J) = (7)3 (3) = 1029

Watch out for

bull probabilities that donrsquot add up to 1

bull donrsquot add probabilities of events if they are not disjoint

bull donrsquot multiply probabilities of events if they are not independent

bull donrsquot confuse disjoint and independent

Rule 7 Conditional Probability

bull A probability that takes into account a given condition

P(A)

B)P(AA)|P(B

P(given)

P(and)A)|P(B

In a class there are 12 boys made up of 8 Seniors and 4 Juniors There are also 8 girls made up of 3 Seniors and 5 Juniors Find the probability of choosing a boy given that he is a Senior

Boy Girl Total

Senior 8 3 11

Junior 4 5 9

Total 12 8 20

P(Boy|Senior) = 811

P(Senior|Boy) = 812 = 23

Probabilities from two way tables

Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359

12) What is the probability that the driver is a student

359195

)( StudentP

Probabilities from two way tables

Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359

13) What is the probability that the driver drives a European car

35945

)( EuropeanP

Probabilities from two way tables

Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359

14) What is the probability that the driver drives an American or Asian car

Disjoint359102212

)(

AsianorAmericanP

Probabilities from two way tables

Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359

15) What is the probability that the driver is staff or drives an Asian car

Disjoint35947102164

)(

AsianorStaffP

Probabilities from two way tables

Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359

16) What is the probability that the driver is staff and drives an Asian car

35947

)( AsianandStaffP

Probabilities from two way tables

Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359

17) If the driver is a student what is the probability that they drive an American car

Condition195107

)|( StudentAmericanP

Probabilities from two way tables

Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359

18) What is the probability that the driver is a student if the driver drives a European car

Condition 4533

)|( EuropeanStudentP

Definition of Independent Events

Two events E and F are independent if and only if

P(F | E) = P(F) or P(E | F) = P(E)

EXAMPLE Illustrating Independent Events

The probability a randomly selected murder victim is male is 07515 The probability a randomly selected murder victim is male given that they are less than 18 years old is 06751

Since P(male) = 07515 and

P(male | lt 18 years old) = 06751

the events ldquomalerdquo and ldquoless than 18 years oldrdquo are not independent In fact knowing the victim is less than 18 years old decreases the probability that the victim is male

I draw one card and look

at it I tell you it is red What is the

probability it is a heart

P( heart | red) =

13P(heart and red) 152

26P(red) 252

Are ldquored cardrdquo and ldquospaderdquo mutually

exclusive Are they

independentA red card canrsquot be a spade so they ARE mutually exclusive

Are ldquored cardrdquo and

ldquoacerdquo mutually

exclusive Are they

independent2 aces are

red cards so they are

NOT mutually exclusive

Are ldquoface cardrdquo and

ldquokingrdquo mutually

exclusive Are they

independentKings are

Face cards so they are

NOT mutually exclusive

ldquoSlump I ainrsquot in no slump I just ainrsquot hittinrdquo

Yogi Berra

• PowerPoint Presentation
• Probability
• Law of Large Numbers
• Basic Rules of Probability
• Slide 5
• Slide 6
• Independent
• Slide 8
• Slide 9
• Slide 10
• Slide 11
• Slide 12
• Slide 13
• Slide 14
• Slide 15
• Slide 16
• Slide 17
• Slide 18
• Slide 19
• Slide 20
• Slide 21
• Slide 22
• Slide 23
• Slide 24
• Slide 25
• Slide 26
• Slide 27
• Slide 28
• Slide 29
• Slide 30
• Slide 31
• Slide 32
• Rule 7 Conditional Probability
• Slide 34
• Probabilities from two way tables
• Slide 36
• Slide 37
• Slide 38
• Slide 39
• Slide 40
• Slide 41
• Slide 42
• Slide 43
• Slide 44
• Slide 45
• Slide 46
• Slide 47
• Slide 48

)( BAP

Independent)( BAP

Yes

)()( BPAP

What does this mean

Given a deck of cards and a die

one card is drawn and the dice is rolled

What is the probability that an ace is drawn and an even is

rolledP(ace and even) = P(ace) P(even) =

4 3 1

52 6 26

)( BAP

Independent)( BAP

Yes No

)()( BPAP )|()( ABPAP

Given a deck of cards two

cards are drawn

without replacement What is the probability

that they are both heartsP(heart and heart) =

)|()( ABPAP

13 12 1

52 51 17

Ex 6) Suppose I will pick two cards from a standard deck without replacement What is the probability that I select two spades

Are the cards independent NO

P(A amp B) = P(A) P(B|A)

Read ldquoprobability of B given that A occursrdquo

P(Spade amp Spade) = 14 1251 = 117

The probability of getting a spade given that a spade has already been drawn

Rule 5 AdditionIf two events E amp F are disjoint

P(E or F) = P(E) + P(F)

If the probability that a randomly selected student is a junior (A) is 2 and the probability that the student is a senior (B) is 5 what is the probability that the student is either a junior or a senior

P(A υ B) = P(A) + P(B) if A and B are disjoint

P(A υ B) = 2 + 5 = 7

bull Two events that have no common outcomes are said to be disjoint or mutually exclusive

A and B are disjoint events

Rule 5 AdditionIf two events E amp F are disjoint P(E or F) = P(E) + P(F)

(General) If two events E amp F are not disjoint

P(E or F) = P(E) + P(F) ndash P(E amp F)Probability of owning a MP3 player 50Probability of owning a computer 90

So the probability of owning a MP3 player or a computer

is 140

Not disjoint events

)( FEP What does this mean

)( FEP Mutually exclusive

)()( FPEP

Yes

Given a deck of cards one

card is drawn What

is the probability

that it is a 3 or a 4

P(3 or 4) = P(3) + P(4) =

4 4 2

52 52 13

)( FEP

)( FEP Mutually exclusive

)()( FPEP

Yes No

)()()( FEPFPEP

Given a deck of cards one

card is drawn What

is the probability that it is an ace or a red

cardP(ace or red) = P(ace) + P(red) ndash P(ace and red) =

52

4

52

26

52

2

52

28+ - =

)( FEP

)( FEP Mutually exclusive

)()( FPEP

Yes No

)()()( FEPFPEP

Independent Yes )()( FPEP

Ex 5)

If P(A) = 045 P(B) = 035 and A amp B are independent find P(A or B)Is A amp B disjoint

If A amp B are disjoint are they independent

Disjoint events do not happen at the same time

So if A occurs can B occur

Disjoint events are dependent

NO independent events cannot be disjoint

P(A or B) = P(A) + P(B) ndash P(A amp B)

How can you find the

probability of A amp B

P(A or B) = 45 + 35 - 45(35) = 06425

If independent

multiply

In a class there are 12 boys made up of 8 Seniors and 4 Juniors There are also 8 girls made up of 3 Seniors and 5 Juniors Find the probability of choosing a boy or a Senior

Note that choosing a boy and choosing a Senior are not disjoint (they can occur simultaneously)

P(boy or a senior) = P(Boy) + P(Senior) ndash P(Senior boy) = 12 11 8 15 3

20 20 20 20 4

Boy Girl Total

Senior 8 3 11

Junior 4 5 9

Total 12 8 20

Rule 6 At least one

The probability that at least one outcome happens is 1 minus the probability that no outcomes happen

P(at least 1) = 1 ndash P(none)

For a sales promotion the manufacturer places winning symbols under the caps of 10 of all Dr Pepper bottles You buy a six-pack What is the probability that you win something

P(at least one winning symbol) =

1 ndash P(no winning symbols) 1 - 96 = 4686

Suppose that 40 of cars in Fort Smith are manufactured in the United States 30 in Japan 10 in Germany and 20 in other countries

If cars are selected at random what is the probability that it is not US made

P(not US made) = 1 ndash P(US made) = 1 - 4 = 6

Suppose that 40 of cars in Fort Smith are manufactured in the United States 30 in Japan 10 in Germany and 20 in other countries

If cars are selected at random what is the probability that it is made in Japan or Germany

P(Japanese or German) = P(Japanese) + P(German) = 3 + 1 = 4

Suppose that 40 of cars in Fort Smith are manufactured in the United States 30 in Japan 10 in Germany and 20 in other countries

If cars are selected at random what is the probability that you see two in a row from Japan

P(2 Japanese in a row) = P(Japanese) and P(Japanese) = P(J) x P(J) = 3 x 3 = 09

Suppose that 40 of cars in Fort Smith are manufactured in the United States 30 in Japan 10 in Germany and 20 in other countries

If cars are selected at random what is the probability that none of three cars came from Germany

P(no Germany in three) = P(not G) x P(not G) x P(not G) = 9 x 9 x 9 = 729

Suppose that 40 of cars in Fort Smith are manufactured in the United States 30 in Japan 10 in Germany and 20 in other countries

If cars are selected at random what is the probability that at least one of three cars is US made

P(at least one US in three) = 1 ndash P(no US in three) = 1 ndash (6)(6)(6) = 784

Suppose that 40 of cars in Fort Smith are manufactured in the United States 30 in Japan 10 in Germany and 20 in other countries

If cars are selected at random what is the probability that the first Japanese car is the fourth one you choose

P(first J is the fourth car) = P(not J) x P(not J) x P(not J) x P(J) = (7)3 (3) = 1029

Watch out for

bull probabilities that donrsquot add up to 1

bull donrsquot add probabilities of events if they are not disjoint

bull donrsquot multiply probabilities of events if they are not independent

bull donrsquot confuse disjoint and independent

Rule 7 Conditional Probability

bull A probability that takes into account a given condition

P(A)

B)P(AA)|P(B

P(given)

P(and)A)|P(B

In a class there are 12 boys made up of 8 Seniors and 4 Juniors There are also 8 girls made up of 3 Seniors and 5 Juniors Find the probability of choosing a boy given that he is a Senior

Boy Girl Total

Senior 8 3 11

Junior 4 5 9

Total 12 8 20

P(Boy|Senior) = 811

P(Senior|Boy) = 812 = 23

Probabilities from two way tables

Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359

12) What is the probability that the driver is a student

359195

)( StudentP

Probabilities from two way tables

Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359

13) What is the probability that the driver drives a European car

35945

)( EuropeanP

Probabilities from two way tables

Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359

14) What is the probability that the driver drives an American or Asian car

Disjoint359102212

)(

AsianorAmericanP

Probabilities from two way tables

Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359

15) What is the probability that the driver is staff or drives an Asian car

Disjoint35947102164

)(

AsianorStaffP

Probabilities from two way tables

Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359

16) What is the probability that the driver is staff and drives an Asian car

35947

)( AsianandStaffP

Probabilities from two way tables

Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359

17) If the driver is a student what is the probability that they drive an American car

Condition195107

)|( StudentAmericanP

Probabilities from two way tables

Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359

18) What is the probability that the driver is a student if the driver drives a European car

Condition 4533

)|( EuropeanStudentP

Definition of Independent Events

Two events E and F are independent if and only if

P(F | E) = P(F) or P(E | F) = P(E)

EXAMPLE Illustrating Independent Events

The probability a randomly selected murder victim is male is 07515 The probability a randomly selected murder victim is male given that they are less than 18 years old is 06751

Since P(male) = 07515 and

P(male | lt 18 years old) = 06751

the events ldquomalerdquo and ldquoless than 18 years oldrdquo are not independent In fact knowing the victim is less than 18 years old decreases the probability that the victim is male

I draw one card and look

at it I tell you it is red What is the

probability it is a heart

P( heart | red) =

13P(heart and red) 152

26P(red) 252

Are ldquored cardrdquo and ldquospaderdquo mutually

exclusive Are they

independentA red card canrsquot be a spade so they ARE mutually exclusive

Are ldquored cardrdquo and

ldquoacerdquo mutually

exclusive Are they

independent2 aces are

red cards so they are

NOT mutually exclusive

Are ldquoface cardrdquo and

ldquokingrdquo mutually

exclusive Are they

independentKings are

Face cards so they are

NOT mutually exclusive

ldquoSlump I ainrsquot in no slump I just ainrsquot hittinrdquo

Yogi Berra

• PowerPoint Presentation
• Probability
• Law of Large Numbers
• Basic Rules of Probability
• Slide 5
• Slide 6
• Independent
• Slide 8
• Slide 9
• Slide 10
• Slide 11
• Slide 12
• Slide 13
• Slide 14
• Slide 15
• Slide 16
• Slide 17
• Slide 18
• Slide 19
• Slide 20
• Slide 21
• Slide 22
• Slide 23
• Slide 24
• Slide 25
• Slide 26
• Slide 27
• Slide 28
• Slide 29
• Slide 30
• Slide 31
• Slide 32
• Rule 7 Conditional Probability
• Slide 34
• Probabilities from two way tables
• Slide 36
• Slide 37
• Slide 38
• Slide 39
• Slide 40
• Slide 41
• Slide 42
• Slide 43
• Slide 44
• Slide 45
• Slide 46
• Slide 47
• Slide 48

Given a deck of cards and a die

one card is drawn and the dice is rolled

What is the probability that an ace is drawn and an even is

rolledP(ace and even) = P(ace) P(even) =

4 3 1

52 6 26

)( BAP

Independent)( BAP

Yes No

)()( BPAP )|()( ABPAP

Given a deck of cards two

cards are drawn

without replacement What is the probability

that they are both heartsP(heart and heart) =

)|()( ABPAP

13 12 1

52 51 17

Ex 6) Suppose I will pick two cards from a standard deck without replacement What is the probability that I select two spades

Are the cards independent NO

P(A amp B) = P(A) P(B|A)

Read ldquoprobability of B given that A occursrdquo

P(Spade amp Spade) = 14 1251 = 117

The probability of getting a spade given that a spade has already been drawn

Rule 5 AdditionIf two events E amp F are disjoint

P(E or F) = P(E) + P(F)

If the probability that a randomly selected student is a junior (A) is 2 and the probability that the student is a senior (B) is 5 what is the probability that the student is either a junior or a senior

P(A υ B) = P(A) + P(B) if A and B are disjoint

P(A υ B) = 2 + 5 = 7

bull Two events that have no common outcomes are said to be disjoint or mutually exclusive

A and B are disjoint events

Rule 5 AdditionIf two events E amp F are disjoint P(E or F) = P(E) + P(F)

(General) If two events E amp F are not disjoint

P(E or F) = P(E) + P(F) ndash P(E amp F)Probability of owning a MP3 player 50Probability of owning a computer 90

So the probability of owning a MP3 player or a computer

is 140

Not disjoint events

)( FEP What does this mean

)( FEP Mutually exclusive

)()( FPEP

Yes

Given a deck of cards one

card is drawn What

is the probability

that it is a 3 or a 4

P(3 or 4) = P(3) + P(4) =

4 4 2

52 52 13

)( FEP

)( FEP Mutually exclusive

)()( FPEP

Yes No

)()()( FEPFPEP

Given a deck of cards one

card is drawn What

is the probability that it is an ace or a red

cardP(ace or red) = P(ace) + P(red) ndash P(ace and red) =

52

4

52

26

52

2

52

28+ - =

)( FEP

)( FEP Mutually exclusive

)()( FPEP

Yes No

)()()( FEPFPEP

Independent Yes )()( FPEP

Ex 5)

If P(A) = 045 P(B) = 035 and A amp B are independent find P(A or B)Is A amp B disjoint

If A amp B are disjoint are they independent

Disjoint events do not happen at the same time

So if A occurs can B occur

Disjoint events are dependent

NO independent events cannot be disjoint

P(A or B) = P(A) + P(B) ndash P(A amp B)

How can you find the

probability of A amp B

P(A or B) = 45 + 35 - 45(35) = 06425

If independent

multiply

In a class there are 12 boys made up of 8 Seniors and 4 Juniors There are also 8 girls made up of 3 Seniors and 5 Juniors Find the probability of choosing a boy or a Senior

Note that choosing a boy and choosing a Senior are not disjoint (they can occur simultaneously)

P(boy or a senior) = P(Boy) + P(Senior) ndash P(Senior boy) = 12 11 8 15 3

20 20 20 20 4

Boy Girl Total

Senior 8 3 11

Junior 4 5 9

Total 12 8 20

Rule 6 At least one

The probability that at least one outcome happens is 1 minus the probability that no outcomes happen

P(at least 1) = 1 ndash P(none)

For a sales promotion the manufacturer places winning symbols under the caps of 10 of all Dr Pepper bottles You buy a six-pack What is the probability that you win something

P(at least one winning symbol) =

1 ndash P(no winning symbols) 1 - 96 = 4686

Suppose that 40 of cars in Fort Smith are manufactured in the United States 30 in Japan 10 in Germany and 20 in other countries

If cars are selected at random what is the probability that it is not US made

P(not US made) = 1 ndash P(US made) = 1 - 4 = 6

Suppose that 40 of cars in Fort Smith are manufactured in the United States 30 in Japan 10 in Germany and 20 in other countries

If cars are selected at random what is the probability that it is made in Japan or Germany

P(Japanese or German) = P(Japanese) + P(German) = 3 + 1 = 4

Suppose that 40 of cars in Fort Smith are manufactured in the United States 30 in Japan 10 in Germany and 20 in other countries

If cars are selected at random what is the probability that you see two in a row from Japan

P(2 Japanese in a row) = P(Japanese) and P(Japanese) = P(J) x P(J) = 3 x 3 = 09

Suppose that 40 of cars in Fort Smith are manufactured in the United States 30 in Japan 10 in Germany and 20 in other countries

If cars are selected at random what is the probability that none of three cars came from Germany

P(no Germany in three) = P(not G) x P(not G) x P(not G) = 9 x 9 x 9 = 729

Suppose that 40 of cars in Fort Smith are manufactured in the United States 30 in Japan 10 in Germany and 20 in other countries

If cars are selected at random what is the probability that at least one of three cars is US made

P(at least one US in three) = 1 ndash P(no US in three) = 1 ndash (6)(6)(6) = 784

Suppose that 40 of cars in Fort Smith are manufactured in the United States 30 in Japan 10 in Germany and 20 in other countries

If cars are selected at random what is the probability that the first Japanese car is the fourth one you choose

P(first J is the fourth car) = P(not J) x P(not J) x P(not J) x P(J) = (7)3 (3) = 1029

Watch out for

bull probabilities that donrsquot add up to 1

bull donrsquot add probabilities of events if they are not disjoint

bull donrsquot multiply probabilities of events if they are not independent

bull donrsquot confuse disjoint and independent

Rule 7 Conditional Probability

bull A probability that takes into account a given condition

P(A)

B)P(AA)|P(B

P(given)

P(and)A)|P(B

In a class there are 12 boys made up of 8 Seniors and 4 Juniors There are also 8 girls made up of 3 Seniors and 5 Juniors Find the probability of choosing a boy given that he is a Senior

Boy Girl Total

Senior 8 3 11

Junior 4 5 9

Total 12 8 20

P(Boy|Senior) = 811

P(Senior|Boy) = 812 = 23

Probabilities from two way tables

Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359

12) What is the probability that the driver is a student

359195

)( StudentP

Probabilities from two way tables

Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359

13) What is the probability that the driver drives a European car

35945

)( EuropeanP

Probabilities from two way tables

Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359

14) What is the probability that the driver drives an American or Asian car

Disjoint359102212

)(

AsianorAmericanP

Probabilities from two way tables

Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359

15) What is the probability that the driver is staff or drives an Asian car

Disjoint35947102164

)(

AsianorStaffP

Probabilities from two way tables

Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359

16) What is the probability that the driver is staff and drives an Asian car

35947

)( AsianandStaffP

Probabilities from two way tables

Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359

17) If the driver is a student what is the probability that they drive an American car

Condition195107

)|( StudentAmericanP

Probabilities from two way tables

Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359

18) What is the probability that the driver is a student if the driver drives a European car

Condition 4533

)|( EuropeanStudentP

Definition of Independent Events

Two events E and F are independent if and only if

P(F | E) = P(F) or P(E | F) = P(E)

EXAMPLE Illustrating Independent Events

The probability a randomly selected murder victim is male is 07515 The probability a randomly selected murder victim is male given that they are less than 18 years old is 06751

Since P(male) = 07515 and

P(male | lt 18 years old) = 06751

the events ldquomalerdquo and ldquoless than 18 years oldrdquo are not independent In fact knowing the victim is less than 18 years old decreases the probability that the victim is male

I draw one card and look

at it I tell you it is red What is the

probability it is a heart

P( heart | red) =

13P(heart and red) 152

26P(red) 252

Are ldquored cardrdquo and ldquospaderdquo mutually

exclusive Are they

independentA red card canrsquot be a spade so they ARE mutually exclusive

Are ldquored cardrdquo and

ldquoacerdquo mutually

exclusive Are they

independent2 aces are

red cards so they are

NOT mutually exclusive

Are ldquoface cardrdquo and

ldquokingrdquo mutually

exclusive Are they

independentKings are

Face cards so they are

NOT mutually exclusive

ldquoSlump I ainrsquot in no slump I just ainrsquot hittinrdquo

Yogi Berra

• PowerPoint Presentation
• Probability
• Law of Large Numbers
• Basic Rules of Probability
• Slide 5
• Slide 6
• Independent
• Slide 8
• Slide 9
• Slide 10
• Slide 11
• Slide 12
• Slide 13
• Slide 14
• Slide 15
• Slide 16
• Slide 17
• Slide 18
• Slide 19
• Slide 20
• Slide 21
• Slide 22
• Slide 23
• Slide 24
• Slide 25
• Slide 26
• Slide 27
• Slide 28
• Slide 29
• Slide 30
• Slide 31
• Slide 32
• Rule 7 Conditional Probability
• Slide 34
• Probabilities from two way tables
• Slide 36
• Slide 37
• Slide 38
• Slide 39
• Slide 40
• Slide 41
• Slide 42
• Slide 43
• Slide 44
• Slide 45
• Slide 46
• Slide 47
• Slide 48

)( BAP

Independent)( BAP

Yes No

)()( BPAP )|()( ABPAP

Given a deck of cards two

cards are drawn

without replacement What is the probability

that they are both heartsP(heart and heart) =

)|()( ABPAP

13 12 1

52 51 17

Ex 6) Suppose I will pick two cards from a standard deck without replacement What is the probability that I select two spades

Are the cards independent NO

P(A amp B) = P(A) P(B|A)

Read ldquoprobability of B given that A occursrdquo

P(Spade amp Spade) = 14 1251 = 117

The probability of getting a spade given that a spade has already been drawn

Rule 5 AdditionIf two events E amp F are disjoint

P(E or F) = P(E) + P(F)

If the probability that a randomly selected student is a junior (A) is 2 and the probability that the student is a senior (B) is 5 what is the probability that the student is either a junior or a senior

P(A υ B) = P(A) + P(B) if A and B are disjoint

P(A υ B) = 2 + 5 = 7

bull Two events that have no common outcomes are said to be disjoint or mutually exclusive

A and B are disjoint events

Rule 5 AdditionIf two events E amp F are disjoint P(E or F) = P(E) + P(F)

(General) If two events E amp F are not disjoint

P(E or F) = P(E) + P(F) ndash P(E amp F)Probability of owning a MP3 player 50Probability of owning a computer 90

So the probability of owning a MP3 player or a computer

is 140

Not disjoint events

)( FEP What does this mean

)( FEP Mutually exclusive

)()( FPEP

Yes

Given a deck of cards one

card is drawn What

is the probability

that it is a 3 or a 4

P(3 or 4) = P(3) + P(4) =

4 4 2

52 52 13

)( FEP

)( FEP Mutually exclusive

)()( FPEP

Yes No

)()()( FEPFPEP

Given a deck of cards one

card is drawn What

is the probability that it is an ace or a red

cardP(ace or red) = P(ace) + P(red) ndash P(ace and red) =

52

4

52

26

52

2

52

28+ - =

)( FEP

)( FEP Mutually exclusive

)()( FPEP

Yes No

)()()( FEPFPEP

Independent Yes )()( FPEP

Ex 5)

If P(A) = 045 P(B) = 035 and A amp B are independent find P(A or B)Is A amp B disjoint

If A amp B are disjoint are they independent

Disjoint events do not happen at the same time

So if A occurs can B occur

Disjoint events are dependent

NO independent events cannot be disjoint

P(A or B) = P(A) + P(B) ndash P(A amp B)

How can you find the

probability of A amp B

P(A or B) = 45 + 35 - 45(35) = 06425

If independent

multiply

In a class there are 12 boys made up of 8 Seniors and 4 Juniors There are also 8 girls made up of 3 Seniors and 5 Juniors Find the probability of choosing a boy or a Senior

Note that choosing a boy and choosing a Senior are not disjoint (they can occur simultaneously)

P(boy or a senior) = P(Boy) + P(Senior) ndash P(Senior boy) = 12 11 8 15 3

20 20 20 20 4

Boy Girl Total

Senior 8 3 11

Junior 4 5 9

Total 12 8 20

Rule 6 At least one

The probability that at least one outcome happens is 1 minus the probability that no outcomes happen

P(at least 1) = 1 ndash P(none)

For a sales promotion the manufacturer places winning symbols under the caps of 10 of all Dr Pepper bottles You buy a six-pack What is the probability that you win something

P(at least one winning symbol) =

1 ndash P(no winning symbols) 1 - 96 = 4686

Suppose that 40 of cars in Fort Smith are manufactured in the United States 30 in Japan 10 in Germany and 20 in other countries

If cars are selected at random what is the probability that it is not US made

P(not US made) = 1 ndash P(US made) = 1 - 4 = 6

Suppose that 40 of cars in Fort Smith are manufactured in the United States 30 in Japan 10 in Germany and 20 in other countries

If cars are selected at random what is the probability that it is made in Japan or Germany

P(Japanese or German) = P(Japanese) + P(German) = 3 + 1 = 4

Suppose that 40 of cars in Fort Smith are manufactured in the United States 30 in Japan 10 in Germany and 20 in other countries

If cars are selected at random what is the probability that you see two in a row from Japan

P(2 Japanese in a row) = P(Japanese) and P(Japanese) = P(J) x P(J) = 3 x 3 = 09

Suppose that 40 of cars in Fort Smith are manufactured in the United States 30 in Japan 10 in Germany and 20 in other countries

If cars are selected at random what is the probability that none of three cars came from Germany

P(no Germany in three) = P(not G) x P(not G) x P(not G) = 9 x 9 x 9 = 729

Suppose that 40 of cars in Fort Smith are manufactured in the United States 30 in Japan 10 in Germany and 20 in other countries

If cars are selected at random what is the probability that at least one of three cars is US made

P(at least one US in three) = 1 ndash P(no US in three) = 1 ndash (6)(6)(6) = 784

Suppose that 40 of cars in Fort Smith are manufactured in the United States 30 in Japan 10 in Germany and 20 in other countries

If cars are selected at random what is the probability that the first Japanese car is the fourth one you choose

P(first J is the fourth car) = P(not J) x P(not J) x P(not J) x P(J) = (7)3 (3) = 1029

Watch out for

bull probabilities that donrsquot add up to 1

bull donrsquot add probabilities of events if they are not disjoint

bull donrsquot multiply probabilities of events if they are not independent

bull donrsquot confuse disjoint and independent

Rule 7 Conditional Probability

bull A probability that takes into account a given condition

P(A)

B)P(AA)|P(B

P(given)

P(and)A)|P(B

In a class there are 12 boys made up of 8 Seniors and 4 Juniors There are also 8 girls made up of 3 Seniors and 5 Juniors Find the probability of choosing a boy given that he is a Senior

Boy Girl Total

Senior 8 3 11

Junior 4 5 9

Total 12 8 20

P(Boy|Senior) = 811

P(Senior|Boy) = 812 = 23

Probabilities from two way tables

Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359

12) What is the probability that the driver is a student

359195

)( StudentP

Probabilities from two way tables

Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359

13) What is the probability that the driver drives a European car

35945

)( EuropeanP

Probabilities from two way tables

Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359

14) What is the probability that the driver drives an American or Asian car

Disjoint359102212

)(

AsianorAmericanP

Probabilities from two way tables

Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359

15) What is the probability that the driver is staff or drives an Asian car

Disjoint35947102164

)(

AsianorStaffP

Probabilities from two way tables

Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359

16) What is the probability that the driver is staff and drives an Asian car

35947

)( AsianandStaffP

Probabilities from two way tables

Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359

17) If the driver is a student what is the probability that they drive an American car

Condition195107

)|( StudentAmericanP

Probabilities from two way tables

Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359

18) What is the probability that the driver is a student if the driver drives a European car

Condition 4533

)|( EuropeanStudentP

Definition of Independent Events

Two events E and F are independent if and only if

P(F | E) = P(F) or P(E | F) = P(E)

EXAMPLE Illustrating Independent Events

The probability a randomly selected murder victim is male is 07515 The probability a randomly selected murder victim is male given that they are less than 18 years old is 06751

Since P(male) = 07515 and

P(male | lt 18 years old) = 06751

the events ldquomalerdquo and ldquoless than 18 years oldrdquo are not independent In fact knowing the victim is less than 18 years old decreases the probability that the victim is male

I draw one card and look

at it I tell you it is red What is the

probability it is a heart

P( heart | red) =

13P(heart and red) 152

26P(red) 252

Are ldquored cardrdquo and ldquospaderdquo mutually

exclusive Are they

independentA red card canrsquot be a spade so they ARE mutually exclusive

Are ldquored cardrdquo and

ldquoacerdquo mutually

exclusive Are they

independent2 aces are

red cards so they are

NOT mutually exclusive

Are ldquoface cardrdquo and

ldquokingrdquo mutually

exclusive Are they

independentKings are

Face cards so they are

NOT mutually exclusive

ldquoSlump I ainrsquot in no slump I just ainrsquot hittinrdquo

Yogi Berra

• PowerPoint Presentation
• Probability
• Law of Large Numbers
• Basic Rules of Probability
• Slide 5
• Slide 6
• Independent
• Slide 8
• Slide 9
• Slide 10
• Slide 11
• Slide 12
• Slide 13
• Slide 14
• Slide 15
• Slide 16
• Slide 17
• Slide 18
• Slide 19
• Slide 20
• Slide 21
• Slide 22
• Slide 23
• Slide 24
• Slide 25
• Slide 26
• Slide 27
• Slide 28
• Slide 29
• Slide 30
• Slide 31
• Slide 32
• Rule 7 Conditional Probability
• Slide 34
• Probabilities from two way tables
• Slide 36
• Slide 37
• Slide 38
• Slide 39
• Slide 40
• Slide 41
• Slide 42
• Slide 43
• Slide 44
• Slide 45
• Slide 46
• Slide 47
• Slide 48

Given a deck of cards two

cards are drawn

without replacement What is the probability

that they are both heartsP(heart and heart) =

)|()( ABPAP

13 12 1

52 51 17

Ex 6) Suppose I will pick two cards from a standard deck without replacement What is the probability that I select two spades

Are the cards independent NO

P(A amp B) = P(A) P(B|A)

Read ldquoprobability of B given that A occursrdquo

P(Spade amp Spade) = 14 1251 = 117

The probability of getting a spade given that a spade has already been drawn

Rule 5 AdditionIf two events E amp F are disjoint

P(E or F) = P(E) + P(F)

If the probability that a randomly selected student is a junior (A) is 2 and the probability that the student is a senior (B) is 5 what is the probability that the student is either a junior or a senior

P(A υ B) = P(A) + P(B) if A and B are disjoint

P(A υ B) = 2 + 5 = 7

bull Two events that have no common outcomes are said to be disjoint or mutually exclusive

A and B are disjoint events

Rule 5 AdditionIf two events E amp F are disjoint P(E or F) = P(E) + P(F)

(General) If two events E amp F are not disjoint

P(E or F) = P(E) + P(F) ndash P(E amp F)Probability of owning a MP3 player 50Probability of owning a computer 90

So the probability of owning a MP3 player or a computer

is 140

Not disjoint events

)( FEP What does this mean

)( FEP Mutually exclusive

)()( FPEP

Yes

Given a deck of cards one

card is drawn What

is the probability

that it is a 3 or a 4

P(3 or 4) = P(3) + P(4) =

4 4 2

52 52 13

)( FEP

)( FEP Mutually exclusive

)()( FPEP

Yes No

)()()( FEPFPEP

Given a deck of cards one

card is drawn What

is the probability that it is an ace or a red

cardP(ace or red) = P(ace) + P(red) ndash P(ace and red) =

52

4

52

26

52

2

52

28+ - =

)( FEP

)( FEP Mutually exclusive

)()( FPEP

Yes No

)()()( FEPFPEP

Independent Yes )()( FPEP

Ex 5)

If P(A) = 045 P(B) = 035 and A amp B are independent find P(A or B)Is A amp B disjoint

If A amp B are disjoint are they independent

Disjoint events do not happen at the same time

So if A occurs can B occur

Disjoint events are dependent

NO independent events cannot be disjoint

P(A or B) = P(A) + P(B) ndash P(A amp B)

How can you find the

probability of A amp B

P(A or B) = 45 + 35 - 45(35) = 06425

If independent

multiply

In a class there are 12 boys made up of 8 Seniors and 4 Juniors There are also 8 girls made up of 3 Seniors and 5 Juniors Find the probability of choosing a boy or a Senior

Note that choosing a boy and choosing a Senior are not disjoint (they can occur simultaneously)

P(boy or a senior) = P(Boy) + P(Senior) ndash P(Senior boy) = 12 11 8 15 3

20 20 20 20 4

Boy Girl Total

Senior 8 3 11

Junior 4 5 9

Total 12 8 20

Rule 6 At least one

The probability that at least one outcome happens is 1 minus the probability that no outcomes happen

P(at least 1) = 1 ndash P(none)

For a sales promotion the manufacturer places winning symbols under the caps of 10 of all Dr Pepper bottles You buy a six-pack What is the probability that you win something

P(at least one winning symbol) =

1 ndash P(no winning symbols) 1 - 96 = 4686

Suppose that 40 of cars in Fort Smith are manufactured in the United States 30 in Japan 10 in Germany and 20 in other countries

If cars are selected at random what is the probability that it is not US made

P(not US made) = 1 ndash P(US made) = 1 - 4 = 6

Suppose that 40 of cars in Fort Smith are manufactured in the United States 30 in Japan 10 in Germany and 20 in other countries

If cars are selected at random what is the probability that it is made in Japan or Germany

P(Japanese or German) = P(Japanese) + P(German) = 3 + 1 = 4

Suppose that 40 of cars in Fort Smith are manufactured in the United States 30 in Japan 10 in Germany and 20 in other countries

If cars are selected at random what is the probability that you see two in a row from Japan

P(2 Japanese in a row) = P(Japanese) and P(Japanese) = P(J) x P(J) = 3 x 3 = 09

Suppose that 40 of cars in Fort Smith are manufactured in the United States 30 in Japan 10 in Germany and 20 in other countries

If cars are selected at random what is the probability that none of three cars came from Germany

P(no Germany in three) = P(not G) x P(not G) x P(not G) = 9 x 9 x 9 = 729

Suppose that 40 of cars in Fort Smith are manufactured in the United States 30 in Japan 10 in Germany and 20 in other countries

If cars are selected at random what is the probability that at least one of three cars is US made

P(at least one US in three) = 1 ndash P(no US in three) = 1 ndash (6)(6)(6) = 784

Suppose that 40 of cars in Fort Smith are manufactured in the United States 30 in Japan 10 in Germany and 20 in other countries

If cars are selected at random what is the probability that the first Japanese car is the fourth one you choose

P(first J is the fourth car) = P(not J) x P(not J) x P(not J) x P(J) = (7)3 (3) = 1029

Watch out for

bull probabilities that donrsquot add up to 1

bull donrsquot add probabilities of events if they are not disjoint

bull donrsquot multiply probabilities of events if they are not independent

bull donrsquot confuse disjoint and independent

Rule 7 Conditional Probability

bull A probability that takes into account a given condition

P(A)

B)P(AA)|P(B

P(given)

P(and)A)|P(B

In a class there are 12 boys made up of 8 Seniors and 4 Juniors There are also 8 girls made up of 3 Seniors and 5 Juniors Find the probability of choosing a boy given that he is a Senior

Boy Girl Total

Senior 8 3 11

Junior 4 5 9

Total 12 8 20

P(Boy|Senior) = 811

P(Senior|Boy) = 812 = 23

Probabilities from two way tables

Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359

12) What is the probability that the driver is a student

359195

)( StudentP

Probabilities from two way tables

Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359

13) What is the probability that the driver drives a European car

35945

)( EuropeanP

Probabilities from two way tables

Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359

14) What is the probability that the driver drives an American or Asian car

Disjoint359102212

)(

AsianorAmericanP

Probabilities from two way tables

Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359

15) What is the probability that the driver is staff or drives an Asian car

Disjoint35947102164

)(

AsianorStaffP

Probabilities from two way tables

Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359

16) What is the probability that the driver is staff and drives an Asian car

35947

)( AsianandStaffP

Probabilities from two way tables

Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359

17) If the driver is a student what is the probability that they drive an American car

Condition195107

)|( StudentAmericanP

Probabilities from two way tables

Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359

18) What is the probability that the driver is a student if the driver drives a European car

Condition 4533

)|( EuropeanStudentP

Definition of Independent Events

Two events E and F are independent if and only if

P(F | E) = P(F) or P(E | F) = P(E)

EXAMPLE Illustrating Independent Events

The probability a randomly selected murder victim is male is 07515 The probability a randomly selected murder victim is male given that they are less than 18 years old is 06751

Since P(male) = 07515 and

P(male | lt 18 years old) = 06751

the events ldquomalerdquo and ldquoless than 18 years oldrdquo are not independent In fact knowing the victim is less than 18 years old decreases the probability that the victim is male

I draw one card and look

at it I tell you it is red What is the

probability it is a heart

P( heart | red) =

13P(heart and red) 152

26P(red) 252

Are ldquored cardrdquo and ldquospaderdquo mutually

exclusive Are they

independentA red card canrsquot be a spade so they ARE mutually exclusive

Are ldquored cardrdquo and

ldquoacerdquo mutually

exclusive Are they

independent2 aces are

red cards so they are

NOT mutually exclusive

Are ldquoface cardrdquo and

ldquokingrdquo mutually

exclusive Are they

independentKings are

Face cards so they are

NOT mutually exclusive

ldquoSlump I ainrsquot in no slump I just ainrsquot hittinrdquo

Yogi Berra

• PowerPoint Presentation
• Probability
• Law of Large Numbers
• Basic Rules of Probability
• Slide 5
• Slide 6
• Independent
• Slide 8
• Slide 9
• Slide 10
• Slide 11
• Slide 12
• Slide 13
• Slide 14
• Slide 15
• Slide 16
• Slide 17
• Slide 18
• Slide 19
• Slide 20
• Slide 21
• Slide 22
• Slide 23
• Slide 24
• Slide 25
• Slide 26
• Slide 27
• Slide 28
• Slide 29
• Slide 30
• Slide 31
• Slide 32
• Rule 7 Conditional Probability
• Slide 34
• Probabilities from two way tables
• Slide 36
• Slide 37
• Slide 38
• Slide 39
• Slide 40
• Slide 41
• Slide 42
• Slide 43
• Slide 44
• Slide 45
• Slide 46
• Slide 47
• Slide 48

Ex 6) Suppose I will pick two cards from a standard deck without replacement What is the probability that I select two spades

Are the cards independent NO

P(A amp B) = P(A) P(B|A)

Read ldquoprobability of B given that A occursrdquo

P(Spade amp Spade) = 14 1251 = 117

The probability of getting a spade given that a spade has already been drawn

Rule 5 AdditionIf two events E amp F are disjoint

P(E or F) = P(E) + P(F)

If the probability that a randomly selected student is a junior (A) is 2 and the probability that the student is a senior (B) is 5 what is the probability that the student is either a junior or a senior

P(A υ B) = P(A) + P(B) if A and B are disjoint

P(A υ B) = 2 + 5 = 7

bull Two events that have no common outcomes are said to be disjoint or mutually exclusive

A and B are disjoint events

Rule 5 AdditionIf two events E amp F are disjoint P(E or F) = P(E) + P(F)

(General) If two events E amp F are not disjoint

P(E or F) = P(E) + P(F) ndash P(E amp F)Probability of owning a MP3 player 50Probability of owning a computer 90

So the probability of owning a MP3 player or a computer

is 140

Not disjoint events

)( FEP What does this mean

)( FEP Mutually exclusive

)()( FPEP

Yes

Given a deck of cards one

card is drawn What

is the probability

that it is a 3 or a 4

P(3 or 4) = P(3) + P(4) =

4 4 2

52 52 13

)( FEP

)( FEP Mutually exclusive

)()( FPEP

Yes No

)()()( FEPFPEP

Given a deck of cards one

card is drawn What

is the probability that it is an ace or a red

cardP(ace or red) = P(ace) + P(red) ndash P(ace and red) =

52

4

52

26

52

2

52

28+ - =

)( FEP

)( FEP Mutually exclusive

)()( FPEP

Yes No

)()()( FEPFPEP

Independent Yes )()( FPEP

Ex 5)

If P(A) = 045 P(B) = 035 and A amp B are independent find P(A or B)Is A amp B disjoint

If A amp B are disjoint are they independent

Disjoint events do not happen at the same time

So if A occurs can B occur

Disjoint events are dependent

NO independent events cannot be disjoint

P(A or B) = P(A) + P(B) ndash P(A amp B)

How can you find the

probability of A amp B

P(A or B) = 45 + 35 - 45(35) = 06425

If independent

multiply

In a class there are 12 boys made up of 8 Seniors and 4 Juniors There are also 8 girls made up of 3 Seniors and 5 Juniors Find the probability of choosing a boy or a Senior

Note that choosing a boy and choosing a Senior are not disjoint (they can occur simultaneously)

P(boy or a senior) = P(Boy) + P(Senior) ndash P(Senior boy) = 12 11 8 15 3

20 20 20 20 4

Boy Girl Total

Senior 8 3 11

Junior 4 5 9

Total 12 8 20

Rule 6 At least one

The probability that at least one outcome happens is 1 minus the probability that no outcomes happen

P(at least 1) = 1 ndash P(none)

For a sales promotion the manufacturer places winning symbols under the caps of 10 of all Dr Pepper bottles You buy a six-pack What is the probability that you win something

P(at least one winning symbol) =

1 ndash P(no winning symbols) 1 - 96 = 4686

Suppose that 40 of cars in Fort Smith are manufactured in the United States 30 in Japan 10 in Germany and 20 in other countries

If cars are selected at random what is the probability that it is not US made

P(not US made) = 1 ndash P(US made) = 1 - 4 = 6

Suppose that 40 of cars in Fort Smith are manufactured in the United States 30 in Japan 10 in Germany and 20 in other countries

If cars are selected at random what is the probability that it is made in Japan or Germany

P(Japanese or German) = P(Japanese) + P(German) = 3 + 1 = 4

Suppose that 40 of cars in Fort Smith are manufactured in the United States 30 in Japan 10 in Germany and 20 in other countries

If cars are selected at random what is the probability that you see two in a row from Japan

P(2 Japanese in a row) = P(Japanese) and P(Japanese) = P(J) x P(J) = 3 x 3 = 09

Suppose that 40 of cars in Fort Smith are manufactured in the United States 30 in Japan 10 in Germany and 20 in other countries

If cars are selected at random what is the probability that none of three cars came from Germany

P(no Germany in three) = P(not G) x P(not G) x P(not G) = 9 x 9 x 9 = 729

Suppose that 40 of cars in Fort Smith are manufactured in the United States 30 in Japan 10 in Germany and 20 in other countries

If cars are selected at random what is the probability that at least one of three cars is US made

P(at least one US in three) = 1 ndash P(no US in three) = 1 ndash (6)(6)(6) = 784

Suppose that 40 of cars in Fort Smith are manufactured in the United States 30 in Japan 10 in Germany and 20 in other countries

If cars are selected at random what is the probability that the first Japanese car is the fourth one you choose

P(first J is the fourth car) = P(not J) x P(not J) x P(not J) x P(J) = (7)3 (3) = 1029

Watch out for

bull probabilities that donrsquot add up to 1

bull donrsquot add probabilities of events if they are not disjoint

bull donrsquot multiply probabilities of events if they are not independent

bull donrsquot confuse disjoint and independent

Rule 7 Conditional Probability

bull A probability that takes into account a given condition

P(A)

B)P(AA)|P(B

P(given)

P(and)A)|P(B

In a class there are 12 boys made up of 8 Seniors and 4 Juniors There are also 8 girls made up of 3 Seniors and 5 Juniors Find the probability of choosing a boy given that he is a Senior

Boy Girl Total

Senior 8 3 11

Junior 4 5 9

Total 12 8 20

P(Boy|Senior) = 811

P(Senior|Boy) = 812 = 23

Probabilities from two way tables

Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359

12) What is the probability that the driver is a student

359195

)( StudentP

Probabilities from two way tables

Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359

13) What is the probability that the driver drives a European car

35945

)( EuropeanP

Probabilities from two way tables

Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359

14) What is the probability that the driver drives an American or Asian car

Disjoint359102212

)(

AsianorAmericanP

Probabilities from two way tables

Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359

15) What is the probability that the driver is staff or drives an Asian car

Disjoint35947102164

)(

AsianorStaffP

Probabilities from two way tables

Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359

16) What is the probability that the driver is staff and drives an Asian car

35947

)( AsianandStaffP

Probabilities from two way tables

Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359

17) If the driver is a student what is the probability that they drive an American car

Condition195107

)|( StudentAmericanP

Probabilities from two way tables

Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359

18) What is the probability that the driver is a student if the driver drives a European car

Condition 4533

)|( EuropeanStudentP

Definition of Independent Events

Two events E and F are independent if and only if

P(F | E) = P(F) or P(E | F) = P(E)

EXAMPLE Illustrating Independent Events

The probability a randomly selected murder victim is male is 07515 The probability a randomly selected murder victim is male given that they are less than 18 years old is 06751

Since P(male) = 07515 and

P(male | lt 18 years old) = 06751

the events ldquomalerdquo and ldquoless than 18 years oldrdquo are not independent In fact knowing the victim is less than 18 years old decreases the probability that the victim is male

I draw one card and look

at it I tell you it is red What is the

probability it is a heart

P( heart | red) =

13P(heart and red) 152

26P(red) 252

Are ldquored cardrdquo and ldquospaderdquo mutually

exclusive Are they

independentA red card canrsquot be a spade so they ARE mutually exclusive

Are ldquored cardrdquo and

ldquoacerdquo mutually

exclusive Are they

independent2 aces are

red cards so they are

NOT mutually exclusive

Are ldquoface cardrdquo and

ldquokingrdquo mutually

exclusive Are they

independentKings are

Face cards so they are

NOT mutually exclusive

ldquoSlump I ainrsquot in no slump I just ainrsquot hittinrdquo

Yogi Berra

• PowerPoint Presentation
• Probability
• Law of Large Numbers
• Basic Rules of Probability
• Slide 5
• Slide 6
• Independent
• Slide 8
• Slide 9
• Slide 10
• Slide 11
• Slide 12
• Slide 13
• Slide 14
• Slide 15
• Slide 16
• Slide 17
• Slide 18
• Slide 19
• Slide 20
• Slide 21
• Slide 22
• Slide 23
• Slide 24
• Slide 25
• Slide 26
• Slide 27
• Slide 28
• Slide 29
• Slide 30
• Slide 31
• Slide 32
• Rule 7 Conditional Probability
• Slide 34
• Probabilities from two way tables
• Slide 36
• Slide 37
• Slide 38
• Slide 39
• Slide 40
• Slide 41
• Slide 42
• Slide 43
• Slide 44
• Slide 45
• Slide 46
• Slide 47
• Slide 48

Rule 5 AdditionIf two events E amp F are disjoint

P(E or F) = P(E) + P(F)

If the probability that a randomly selected student is a junior (A) is 2 and the probability that the student is a senior (B) is 5 what is the probability that the student is either a junior or a senior

P(A υ B) = P(A) + P(B) if A and B are disjoint

P(A υ B) = 2 + 5 = 7

bull Two events that have no common outcomes are said to be disjoint or mutually exclusive

A and B are disjoint events

Rule 5 AdditionIf two events E amp F are disjoint P(E or F) = P(E) + P(F)

(General) If two events E amp F are not disjoint

P(E or F) = P(E) + P(F) ndash P(E amp F)Probability of owning a MP3 player 50Probability of owning a computer 90

So the probability of owning a MP3 player or a computer

is 140

Not disjoint events

)( FEP What does this mean

)( FEP Mutually exclusive

)()( FPEP

Yes

Given a deck of cards one

card is drawn What

is the probability

that it is a 3 or a 4

P(3 or 4) = P(3) + P(4) =

4 4 2

52 52 13

)( FEP

)( FEP Mutually exclusive

)()( FPEP

Yes No

)()()( FEPFPEP

Given a deck of cards one

card is drawn What

is the probability that it is an ace or a red

cardP(ace or red) = P(ace) + P(red) ndash P(ace and red) =

52

4

52

26

52

2

52

28+ - =

)( FEP

)( FEP Mutually exclusive

)()( FPEP

Yes No

)()()( FEPFPEP

Independent Yes )()( FPEP

Ex 5)

If P(A) = 045 P(B) = 035 and A amp B are independent find P(A or B)Is A amp B disjoint

If A amp B are disjoint are they independent

Disjoint events do not happen at the same time

So if A occurs can B occur

Disjoint events are dependent

NO independent events cannot be disjoint

P(A or B) = P(A) + P(B) ndash P(A amp B)

How can you find the

probability of A amp B

P(A or B) = 45 + 35 - 45(35) = 06425

If independent

multiply

In a class there are 12 boys made up of 8 Seniors and 4 Juniors There are also 8 girls made up of 3 Seniors and 5 Juniors Find the probability of choosing a boy or a Senior

Note that choosing a boy and choosing a Senior are not disjoint (they can occur simultaneously)

P(boy or a senior) = P(Boy) + P(Senior) ndash P(Senior boy) = 12 11 8 15 3

20 20 20 20 4

Boy Girl Total

Senior 8 3 11

Junior 4 5 9

Total 12 8 20

Rule 6 At least one

The probability that at least one outcome happens is 1 minus the probability that no outcomes happen

P(at least 1) = 1 ndash P(none)

For a sales promotion the manufacturer places winning symbols under the caps of 10 of all Dr Pepper bottles You buy a six-pack What is the probability that you win something

P(at least one winning symbol) =

1 ndash P(no winning symbols) 1 - 96 = 4686

Suppose that 40 of cars in Fort Smith are manufactured in the United States 30 in Japan 10 in Germany and 20 in other countries

If cars are selected at random what is the probability that it is not US made

P(not US made) = 1 ndash P(US made) = 1 - 4 = 6

Suppose that 40 of cars in Fort Smith are manufactured in the United States 30 in Japan 10 in Germany and 20 in other countries

If cars are selected at random what is the probability that it is made in Japan or Germany

P(Japanese or German) = P(Japanese) + P(German) = 3 + 1 = 4

Suppose that 40 of cars in Fort Smith are manufactured in the United States 30 in Japan 10 in Germany and 20 in other countries

If cars are selected at random what is the probability that you see two in a row from Japan

P(2 Japanese in a row) = P(Japanese) and P(Japanese) = P(J) x P(J) = 3 x 3 = 09

Suppose that 40 of cars in Fort Smith are manufactured in the United States 30 in Japan 10 in Germany and 20 in other countries

If cars are selected at random what is the probability that none of three cars came from Germany

P(no Germany in three) = P(not G) x P(not G) x P(not G) = 9 x 9 x 9 = 729

Suppose that 40 of cars in Fort Smith are manufactured in the United States 30 in Japan 10 in Germany and 20 in other countries

If cars are selected at random what is the probability that at least one of three cars is US made

P(at least one US in three) = 1 ndash P(no US in three) = 1 ndash (6)(6)(6) = 784

Suppose that 40 of cars in Fort Smith are manufactured in the United States 30 in Japan 10 in Germany and 20 in other countries

If cars are selected at random what is the probability that the first Japanese car is the fourth one you choose

P(first J is the fourth car) = P(not J) x P(not J) x P(not J) x P(J) = (7)3 (3) = 1029

Watch out for

bull probabilities that donrsquot add up to 1

bull donrsquot add probabilities of events if they are not disjoint

bull donrsquot multiply probabilities of events if they are not independent

bull donrsquot confuse disjoint and independent

Rule 7 Conditional Probability

bull A probability that takes into account a given condition

P(A)

B)P(AA)|P(B

P(given)

P(and)A)|P(B

In a class there are 12 boys made up of 8 Seniors and 4 Juniors There are also 8 girls made up of 3 Seniors and 5 Juniors Find the probability of choosing a boy given that he is a Senior

Boy Girl Total

Senior 8 3 11

Junior 4 5 9

Total 12 8 20

P(Boy|Senior) = 811

P(Senior|Boy) = 812 = 23

Probabilities from two way tables

Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359

12) What is the probability that the driver is a student

359195

)( StudentP

Probabilities from two way tables

Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359

13) What is the probability that the driver drives a European car

35945

)( EuropeanP

Probabilities from two way tables

Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359

14) What is the probability that the driver drives an American or Asian car

Disjoint359102212

)(

AsianorAmericanP

Probabilities from two way tables

Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359

15) What is the probability that the driver is staff or drives an Asian car

Disjoint35947102164

)(

AsianorStaffP

Probabilities from two way tables

Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359

16) What is the probability that the driver is staff and drives an Asian car

35947

)( AsianandStaffP

Probabilities from two way tables

Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359

17) If the driver is a student what is the probability that they drive an American car

Condition195107

)|( StudentAmericanP

Probabilities from two way tables

Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359

18) What is the probability that the driver is a student if the driver drives a European car

Condition 4533

)|( EuropeanStudentP

Definition of Independent Events

Two events E and F are independent if and only if

P(F | E) = P(F) or P(E | F) = P(E)

EXAMPLE Illustrating Independent Events

The probability a randomly selected murder victim is male is 07515 The probability a randomly selected murder victim is male given that they are less than 18 years old is 06751

Since P(male) = 07515 and

P(male | lt 18 years old) = 06751

the events ldquomalerdquo and ldquoless than 18 years oldrdquo are not independent In fact knowing the victim is less than 18 years old decreases the probability that the victim is male

I draw one card and look

at it I tell you it is red What is the

probability it is a heart

P( heart | red) =

13P(heart and red) 152

26P(red) 252

Are ldquored cardrdquo and ldquospaderdquo mutually

exclusive Are they

independentA red card canrsquot be a spade so they ARE mutually exclusive

Are ldquored cardrdquo and

ldquoacerdquo mutually

exclusive Are they

independent2 aces are

red cards so they are

NOT mutually exclusive

Are ldquoface cardrdquo and

ldquokingrdquo mutually

exclusive Are they

independentKings are

Face cards so they are

NOT mutually exclusive

ldquoSlump I ainrsquot in no slump I just ainrsquot hittinrdquo

Yogi Berra

• PowerPoint Presentation
• Probability
• Law of Large Numbers
• Basic Rules of Probability
• Slide 5
• Slide 6
• Independent
• Slide 8
• Slide 9
• Slide 10
• Slide 11
• Slide 12
• Slide 13
• Slide 14
• Slide 15
• Slide 16
• Slide 17
• Slide 18
• Slide 19
• Slide 20
• Slide 21
• Slide 22
• Slide 23
• Slide 24
• Slide 25
• Slide 26
• Slide 27
• Slide 28
• Slide 29
• Slide 30
• Slide 31
• Slide 32
• Rule 7 Conditional Probability
• Slide 34
• Probabilities from two way tables
• Slide 36
• Slide 37
• Slide 38
• Slide 39
• Slide 40
• Slide 41
• Slide 42
• Slide 43
• Slide 44
• Slide 45
• Slide 46
• Slide 47
• Slide 48

bull Two events that have no common outcomes are said to be disjoint or mutually exclusive

A and B are disjoint events

Rule 5 AdditionIf two events E amp F are disjoint P(E or F) = P(E) + P(F)

(General) If two events E amp F are not disjoint

P(E or F) = P(E) + P(F) ndash P(E amp F)Probability of owning a MP3 player 50Probability of owning a computer 90

So the probability of owning a MP3 player or a computer

is 140

Not disjoint events

)( FEP What does this mean

)( FEP Mutually exclusive

)()( FPEP

Yes

Given a deck of cards one

card is drawn What

is the probability

that it is a 3 or a 4

P(3 or 4) = P(3) + P(4) =

4 4 2

52 52 13

)( FEP

)( FEP Mutually exclusive

)()( FPEP

Yes No

)()()( FEPFPEP

Given a deck of cards one

card is drawn What

is the probability that it is an ace or a red

cardP(ace or red) = P(ace) + P(red) ndash P(ace and red) =

52

4

52

26

52

2

52

28+ - =

)( FEP

)( FEP Mutually exclusive

)()( FPEP

Yes No

)()()( FEPFPEP

Independent Yes )()( FPEP

Ex 5)

If P(A) = 045 P(B) = 035 and A amp B are independent find P(A or B)Is A amp B disjoint

If A amp B are disjoint are they independent

Disjoint events do not happen at the same time

So if A occurs can B occur

Disjoint events are dependent

NO independent events cannot be disjoint

P(A or B) = P(A) + P(B) ndash P(A amp B)

How can you find the

probability of A amp B

P(A or B) = 45 + 35 - 45(35) = 06425

If independent

multiply

In a class there are 12 boys made up of 8 Seniors and 4 Juniors There are also 8 girls made up of 3 Seniors and 5 Juniors Find the probability of choosing a boy or a Senior

Note that choosing a boy and choosing a Senior are not disjoint (they can occur simultaneously)

P(boy or a senior) = P(Boy) + P(Senior) ndash P(Senior boy) = 12 11 8 15 3

20 20 20 20 4

Boy Girl Total

Senior 8 3 11

Junior 4 5 9

Total 12 8 20

Rule 6 At least one

The probability that at least one outcome happens is 1 minus the probability that no outcomes happen

P(at least 1) = 1 ndash P(none)

For a sales promotion the manufacturer places winning symbols under the caps of 10 of all Dr Pepper bottles You buy a six-pack What is the probability that you win something

P(at least one winning symbol) =

1 ndash P(no winning symbols) 1 - 96 = 4686

Suppose that 40 of cars in Fort Smith are manufactured in the United States 30 in Japan 10 in Germany and 20 in other countries

If cars are selected at random what is the probability that it is not US made

P(not US made) = 1 ndash P(US made) = 1 - 4 = 6

Suppose that 40 of cars in Fort Smith are manufactured in the United States 30 in Japan 10 in Germany and 20 in other countries

If cars are selected at random what is the probability that it is made in Japan or Germany

P(Japanese or German) = P(Japanese) + P(German) = 3 + 1 = 4

Suppose that 40 of cars in Fort Smith are manufactured in the United States 30 in Japan 10 in Germany and 20 in other countries

If cars are selected at random what is the probability that you see two in a row from Japan

P(2 Japanese in a row) = P(Japanese) and P(Japanese) = P(J) x P(J) = 3 x 3 = 09

Suppose that 40 of cars in Fort Smith are manufactured in the United States 30 in Japan 10 in Germany and 20 in other countries

If cars are selected at random what is the probability that none of three cars came from Germany

P(no Germany in three) = P(not G) x P(not G) x P(not G) = 9 x 9 x 9 = 729

Suppose that 40 of cars in Fort Smith are manufactured in the United States 30 in Japan 10 in Germany and 20 in other countries

If cars are selected at random what is the probability that at least one of three cars is US made

P(at least one US in three) = 1 ndash P(no US in three) = 1 ndash (6)(6)(6) = 784

Suppose that 40 of cars in Fort Smith are manufactured in the United States 30 in Japan 10 in Germany and 20 in other countries

If cars are selected at random what is the probability that the first Japanese car is the fourth one you choose

P(first J is the fourth car) = P(not J) x P(not J) x P(not J) x P(J) = (7)3 (3) = 1029

Watch out for

bull probabilities that donrsquot add up to 1

bull donrsquot add probabilities of events if they are not disjoint

bull donrsquot multiply probabilities of events if they are not independent

bull donrsquot confuse disjoint and independent

Rule 7 Conditional Probability

bull A probability that takes into account a given condition

P(A)

B)P(AA)|P(B

P(given)

P(and)A)|P(B

In a class there are 12 boys made up of 8 Seniors and 4 Juniors There are also 8 girls made up of 3 Seniors and 5 Juniors Find the probability of choosing a boy given that he is a Senior

Boy Girl Total

Senior 8 3 11

Junior 4 5 9

Total 12 8 20

P(Boy|Senior) = 811

P(Senior|Boy) = 812 = 23

Probabilities from two way tables

Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359

12) What is the probability that the driver is a student

359195

)( StudentP

Probabilities from two way tables

Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359

13) What is the probability that the driver drives a European car

35945

)( EuropeanP

Probabilities from two way tables

Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359

14) What is the probability that the driver drives an American or Asian car

Disjoint359102212

)(

AsianorAmericanP

Probabilities from two way tables

Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359

15) What is the probability that the driver is staff or drives an Asian car

Disjoint35947102164

)(

AsianorStaffP

Probabilities from two way tables

Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359

16) What is the probability that the driver is staff and drives an Asian car

35947

)( AsianandStaffP

Probabilities from two way tables

Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359

17) If the driver is a student what is the probability that they drive an American car

Condition195107

)|( StudentAmericanP

Probabilities from two way tables

Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359

18) What is the probability that the driver is a student if the driver drives a European car

Condition 4533

)|( EuropeanStudentP

Definition of Independent Events

Two events E and F are independent if and only if

P(F | E) = P(F) or P(E | F) = P(E)

EXAMPLE Illustrating Independent Events

The probability a randomly selected murder victim is male is 07515 The probability a randomly selected murder victim is male given that they are less than 18 years old is 06751

Since P(male) = 07515 and

P(male | lt 18 years old) = 06751

the events ldquomalerdquo and ldquoless than 18 years oldrdquo are not independent In fact knowing the victim is less than 18 years old decreases the probability that the victim is male

I draw one card and look

at it I tell you it is red What is the

probability it is a heart

P( heart | red) =

13P(heart and red) 152

26P(red) 252

Are ldquored cardrdquo and ldquospaderdquo mutually

exclusive Are they

independentA red card canrsquot be a spade so they ARE mutually exclusive

Are ldquored cardrdquo and

ldquoacerdquo mutually

exclusive Are they

independent2 aces are

red cards so they are

NOT mutually exclusive

Are ldquoface cardrdquo and

ldquokingrdquo mutually

exclusive Are they

independentKings are

Face cards so they are

NOT mutually exclusive

ldquoSlump I ainrsquot in no slump I just ainrsquot hittinrdquo

Yogi Berra

• PowerPoint Presentation
• Probability
• Law of Large Numbers
• Basic Rules of Probability
• Slide 5
• Slide 6
• Independent
• Slide 8
• Slide 9
• Slide 10
• Slide 11
• Slide 12
• Slide 13
• Slide 14
• Slide 15
• Slide 16
• Slide 17
• Slide 18
• Slide 19
• Slide 20
• Slide 21
• Slide 22
• Slide 23
• Slide 24
• Slide 25
• Slide 26
• Slide 27
• Slide 28
• Slide 29
• Slide 30
• Slide 31
• Slide 32
• Rule 7 Conditional Probability
• Slide 34
• Probabilities from two way tables
• Slide 36
• Slide 37
• Slide 38
• Slide 39
• Slide 40
• Slide 41
• Slide 42
• Slide 43
• Slide 44
• Slide 45
• Slide 46
• Slide 47
• Slide 48

Rule 5 AdditionIf two events E amp F are disjoint P(E or F) = P(E) + P(F)

(General) If two events E amp F are not disjoint

P(E or F) = P(E) + P(F) ndash P(E amp F)Probability of owning a MP3 player 50Probability of owning a computer 90

So the probability of owning a MP3 player or a computer

is 140

Not disjoint events

)( FEP What does this mean

)( FEP Mutually exclusive

)()( FPEP

Yes

Given a deck of cards one

card is drawn What

is the probability

that it is a 3 or a 4

P(3 or 4) = P(3) + P(4) =

4 4 2

52 52 13

)( FEP

)( FEP Mutually exclusive

)()( FPEP

Yes No

)()()( FEPFPEP

Given a deck of cards one

card is drawn What

is the probability that it is an ace or a red

cardP(ace or red) = P(ace) + P(red) ndash P(ace and red) =

52

4

52

26

52

2

52

28+ - =

)( FEP

)( FEP Mutually exclusive

)()( FPEP

Yes No

)()()( FEPFPEP

Independent Yes )()( FPEP

Ex 5)

If P(A) = 045 P(B) = 035 and A amp B are independent find P(A or B)Is A amp B disjoint

If A amp B are disjoint are they independent

Disjoint events do not happen at the same time

So if A occurs can B occur

Disjoint events are dependent

NO independent events cannot be disjoint

P(A or B) = P(A) + P(B) ndash P(A amp B)

How can you find the

probability of A amp B

P(A or B) = 45 + 35 - 45(35) = 06425

If independent

multiply

In a class there are 12 boys made up of 8 Seniors and 4 Juniors There are also 8 girls made up of 3 Seniors and 5 Juniors Find the probability of choosing a boy or a Senior

Note that choosing a boy and choosing a Senior are not disjoint (they can occur simultaneously)

P(boy or a senior) = P(Boy) + P(Senior) ndash P(Senior boy) = 12 11 8 15 3

20 20 20 20 4

Boy Girl Total

Senior 8 3 11

Junior 4 5 9

Total 12 8 20

Rule 6 At least one

The probability that at least one outcome happens is 1 minus the probability that no outcomes happen

P(at least 1) = 1 ndash P(none)

For a sales promotion the manufacturer places winning symbols under the caps of 10 of all Dr Pepper bottles You buy a six-pack What is the probability that you win something

P(at least one winning symbol) =

1 ndash P(no winning symbols) 1 - 96 = 4686

Suppose that 40 of cars in Fort Smith are manufactured in the United States 30 in Japan 10 in Germany and 20 in other countries

If cars are selected at random what is the probability that it is not US made

P(not US made) = 1 ndash P(US made) = 1 - 4 = 6

Suppose that 40 of cars in Fort Smith are manufactured in the United States 30 in Japan 10 in Germany and 20 in other countries

If cars are selected at random what is the probability that it is made in Japan or Germany

P(Japanese or German) = P(Japanese) + P(German) = 3 + 1 = 4

Suppose that 40 of cars in Fort Smith are manufactured in the United States 30 in Japan 10 in Germany and 20 in other countries

If cars are selected at random what is the probability that you see two in a row from Japan

P(2 Japanese in a row) = P(Japanese) and P(Japanese) = P(J) x P(J) = 3 x 3 = 09

Suppose that 40 of cars in Fort Smith are manufactured in the United States 30 in Japan 10 in Germany and 20 in other countries

If cars are selected at random what is the probability that none of three cars came from Germany

P(no Germany in three) = P(not G) x P(not G) x P(not G) = 9 x 9 x 9 = 729

Suppose that 40 of cars in Fort Smith are manufactured in the United States 30 in Japan 10 in Germany and 20 in other countries

If cars are selected at random what is the probability that at least one of three cars is US made

P(at least one US in three) = 1 ndash P(no US in three) = 1 ndash (6)(6)(6) = 784

Suppose that 40 of cars in Fort Smith are manufactured in the United States 30 in Japan 10 in Germany and 20 in other countries

If cars are selected at random what is the probability that the first Japanese car is the fourth one you choose

P(first J is the fourth car) = P(not J) x P(not J) x P(not J) x P(J) = (7)3 (3) = 1029

Watch out for

bull probabilities that donrsquot add up to 1

bull donrsquot add probabilities of events if they are not disjoint

bull donrsquot multiply probabilities of events if they are not independent

bull donrsquot confuse disjoint and independent

Rule 7 Conditional Probability

bull A probability that takes into account a given condition

P(A)

B)P(AA)|P(B

P(given)

P(and)A)|P(B

In a class there are 12 boys made up of 8 Seniors and 4 Juniors There are also 8 girls made up of 3 Seniors and 5 Juniors Find the probability of choosing a boy given that he is a Senior

Boy Girl Total

Senior 8 3 11

Junior 4 5 9

Total 12 8 20

P(Boy|Senior) = 811

P(Senior|Boy) = 812 = 23

Probabilities from two way tables

Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359

12) What is the probability that the driver is a student

359195

)( StudentP

Probabilities from two way tables

Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359

13) What is the probability that the driver drives a European car

35945

)( EuropeanP

Probabilities from two way tables

Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359

14) What is the probability that the driver drives an American or Asian car

Disjoint359102212

)(

AsianorAmericanP

Probabilities from two way tables

Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359

15) What is the probability that the driver is staff or drives an Asian car

Disjoint35947102164

)(

AsianorStaffP

Probabilities from two way tables

Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359

16) What is the probability that the driver is staff and drives an Asian car

35947

)( AsianandStaffP

Probabilities from two way tables

Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359

17) If the driver is a student what is the probability that they drive an American car

Condition195107

)|( StudentAmericanP

Probabilities from two way tables

Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359

18) What is the probability that the driver is a student if the driver drives a European car

Condition 4533

)|( EuropeanStudentP

Definition of Independent Events

Two events E and F are independent if and only if

P(F | E) = P(F) or P(E | F) = P(E)

EXAMPLE Illustrating Independent Events

The probability a randomly selected murder victim is male is 07515 The probability a randomly selected murder victim is male given that they are less than 18 years old is 06751

Since P(male) = 07515 and

P(male | lt 18 years old) = 06751

the events ldquomalerdquo and ldquoless than 18 years oldrdquo are not independent In fact knowing the victim is less than 18 years old decreases the probability that the victim is male

I draw one card and look

at it I tell you it is red What is the

probability it is a heart

P( heart | red) =

13P(heart and red) 152

26P(red) 252

Are ldquored cardrdquo and ldquospaderdquo mutually

exclusive Are they

independentA red card canrsquot be a spade so they ARE mutually exclusive

Are ldquored cardrdquo and

ldquoacerdquo mutually

exclusive Are they

independent2 aces are

red cards so they are

NOT mutually exclusive

Are ldquoface cardrdquo and

ldquokingrdquo mutually

exclusive Are they

independentKings are

Face cards so they are

NOT mutually exclusive

ldquoSlump I ainrsquot in no slump I just ainrsquot hittinrdquo

Yogi Berra

• PowerPoint Presentation
• Probability
• Law of Large Numbers
• Basic Rules of Probability
• Slide 5
• Slide 6
• Independent
• Slide 8
• Slide 9
• Slide 10
• Slide 11
• Slide 12
• Slide 13
• Slide 14
• Slide 15
• Slide 16
• Slide 17
• Slide 18
• Slide 19
• Slide 20
• Slide 21
• Slide 22
• Slide 23
• Slide 24
• Slide 25
• Slide 26
• Slide 27
• Slide 28
• Slide 29
• Slide 30
• Slide 31
• Slide 32
• Rule 7 Conditional Probability
• Slide 34
• Probabilities from two way tables
• Slide 36
• Slide 37
• Slide 38
• Slide 39
• Slide 40
• Slide 41
• Slide 42
• Slide 43
• Slide 44
• Slide 45
• Slide 46
• Slide 47
• Slide 48

)( FEP What does this mean

)( FEP Mutually exclusive

)()( FPEP

Yes

Given a deck of cards one

card is drawn What

is the probability

that it is a 3 or a 4

P(3 or 4) = P(3) + P(4) =

4 4 2

52 52 13

)( FEP

)( FEP Mutually exclusive

)()( FPEP

Yes No

)()()( FEPFPEP

Given a deck of cards one

card is drawn What

is the probability that it is an ace or a red

cardP(ace or red) = P(ace) + P(red) ndash P(ace and red) =

52

4

52

26

52

2

52

28+ - =

)( FEP

)( FEP Mutually exclusive

)()( FPEP

Yes No

)()()( FEPFPEP

Independent Yes )()( FPEP

Ex 5)

If P(A) = 045 P(B) = 035 and A amp B are independent find P(A or B)Is A amp B disjoint

If A amp B are disjoint are they independent

Disjoint events do not happen at the same time

So if A occurs can B occur

Disjoint events are dependent

NO independent events cannot be disjoint

P(A or B) = P(A) + P(B) ndash P(A amp B)

How can you find the

probability of A amp B

P(A or B) = 45 + 35 - 45(35) = 06425

If independent

multiply

In a class there are 12 boys made up of 8 Seniors and 4 Juniors There are also 8 girls made up of 3 Seniors and 5 Juniors Find the probability of choosing a boy or a Senior

Note that choosing a boy and choosing a Senior are not disjoint (they can occur simultaneously)

P(boy or a senior) = P(Boy) + P(Senior) ndash P(Senior boy) = 12 11 8 15 3

20 20 20 20 4

Boy Girl Total

Senior 8 3 11

Junior 4 5 9

Total 12 8 20

Rule 6 At least one

The probability that at least one outcome happens is 1 minus the probability that no outcomes happen

P(at least 1) = 1 ndash P(none)

For a sales promotion the manufacturer places winning symbols under the caps of 10 of all Dr Pepper bottles You buy a six-pack What is the probability that you win something

P(at least one winning symbol) =

1 ndash P(no winning symbols) 1 - 96 = 4686

Suppose that 40 of cars in Fort Smith are manufactured in the United States 30 in Japan 10 in Germany and 20 in other countries

If cars are selected at random what is the probability that it is not US made

P(not US made) = 1 ndash P(US made) = 1 - 4 = 6

Suppose that 40 of cars in Fort Smith are manufactured in the United States 30 in Japan 10 in Germany and 20 in other countries

If cars are selected at random what is the probability that it is made in Japan or Germany

P(Japanese or German) = P(Japanese) + P(German) = 3 + 1 = 4

Suppose that 40 of cars in Fort Smith are manufactured in the United States 30 in Japan 10 in Germany and 20 in other countries

If cars are selected at random what is the probability that you see two in a row from Japan

P(2 Japanese in a row) = P(Japanese) and P(Japanese) = P(J) x P(J) = 3 x 3 = 09

Suppose that 40 of cars in Fort Smith are manufactured in the United States 30 in Japan 10 in Germany and 20 in other countries

If cars are selected at random what is the probability that none of three cars came from Germany

P(no Germany in three) = P(not G) x P(not G) x P(not G) = 9 x 9 x 9 = 729

Suppose that 40 of cars in Fort Smith are manufactured in the United States 30 in Japan 10 in Germany and 20 in other countries

If cars are selected at random what is the probability that at least one of three cars is US made

P(at least one US in three) = 1 ndash P(no US in three) = 1 ndash (6)(6)(6) = 784

Suppose that 40 of cars in Fort Smith are manufactured in the United States 30 in Japan 10 in Germany and 20 in other countries

If cars are selected at random what is the probability that the first Japanese car is the fourth one you choose

P(first J is the fourth car) = P(not J) x P(not J) x P(not J) x P(J) = (7)3 (3) = 1029

Watch out for

bull probabilities that donrsquot add up to 1

bull donrsquot add probabilities of events if they are not disjoint

bull donrsquot multiply probabilities of events if they are not independent

bull donrsquot confuse disjoint and independent

Rule 7 Conditional Probability

bull A probability that takes into account a given condition

P(A)

B)P(AA)|P(B

P(given)

P(and)A)|P(B

In a class there are 12 boys made up of 8 Seniors and 4 Juniors There are also 8 girls made up of 3 Seniors and 5 Juniors Find the probability of choosing a boy given that he is a Senior

Boy Girl Total

Senior 8 3 11

Junior 4 5 9

Total 12 8 20

P(Boy|Senior) = 811

P(Senior|Boy) = 812 = 23

Probabilities from two way tables

Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359

12) What is the probability that the driver is a student

359195

)( StudentP

Probabilities from two way tables

Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359

13) What is the probability that the driver drives a European car

35945

)( EuropeanP

Probabilities from two way tables

Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359

14) What is the probability that the driver drives an American or Asian car

Disjoint359102212

)(

AsianorAmericanP

Probabilities from two way tables

Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359

15) What is the probability that the driver is staff or drives an Asian car

Disjoint35947102164

)(

AsianorStaffP

Probabilities from two way tables

Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359

16) What is the probability that the driver is staff and drives an Asian car

35947

)( AsianandStaffP

Probabilities from two way tables

Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359

17) If the driver is a student what is the probability that they drive an American car

Condition195107

)|( StudentAmericanP

Probabilities from two way tables

Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359

18) What is the probability that the driver is a student if the driver drives a European car

Condition 4533

)|( EuropeanStudentP

Definition of Independent Events

Two events E and F are independent if and only if

P(F | E) = P(F) or P(E | F) = P(E)

EXAMPLE Illustrating Independent Events

The probability a randomly selected murder victim is male is 07515 The probability a randomly selected murder victim is male given that they are less than 18 years old is 06751

Since P(male) = 07515 and

P(male | lt 18 years old) = 06751

the events ldquomalerdquo and ldquoless than 18 years oldrdquo are not independent In fact knowing the victim is less than 18 years old decreases the probability that the victim is male

I draw one card and look

at it I tell you it is red What is the

probability it is a heart

P( heart | red) =

13P(heart and red) 152

26P(red) 252

Are ldquored cardrdquo and ldquospaderdquo mutually

exclusive Are they

independentA red card canrsquot be a spade so they ARE mutually exclusive

Are ldquored cardrdquo and

ldquoacerdquo mutually

exclusive Are they

independent2 aces are

red cards so they are

NOT mutually exclusive

Are ldquoface cardrdquo and

ldquokingrdquo mutually

exclusive Are they

independentKings are

Face cards so they are

NOT mutually exclusive

ldquoSlump I ainrsquot in no slump I just ainrsquot hittinrdquo

Yogi Berra

• PowerPoint Presentation
• Probability
• Law of Large Numbers
• Basic Rules of Probability
• Slide 5
• Slide 6
• Independent
• Slide 8
• Slide 9
• Slide 10
• Slide 11
• Slide 12
• Slide 13
• Slide 14
• Slide 15
• Slide 16
• Slide 17
• Slide 18
• Slide 19
• Slide 20
• Slide 21
• Slide 22
• Slide 23
• Slide 24
• Slide 25
• Slide 26
• Slide 27
• Slide 28
• Slide 29
• Slide 30
• Slide 31
• Slide 32
• Rule 7 Conditional Probability
• Slide 34
• Probabilities from two way tables
• Slide 36
• Slide 37
• Slide 38
• Slide 39
• Slide 40
• Slide 41
• Slide 42
• Slide 43
• Slide 44
• Slide 45
• Slide 46
• Slide 47
• Slide 48

Given a deck of cards one

card is drawn What

is the probability

that it is a 3 or a 4

P(3 or 4) = P(3) + P(4) =

4 4 2

52 52 13

)( FEP

)( FEP Mutually exclusive

)()( FPEP

Yes No

)()()( FEPFPEP

Given a deck of cards one

card is drawn What

is the probability that it is an ace or a red

cardP(ace or red) = P(ace) + P(red) ndash P(ace and red) =

52

4

52

26

52

2

52

28+ - =

)( FEP

)( FEP Mutually exclusive

)()( FPEP

Yes No

)()()( FEPFPEP

Independent Yes )()( FPEP

Ex 5)

If P(A) = 045 P(B) = 035 and A amp B are independent find P(A or B)Is A amp B disjoint

If A amp B are disjoint are they independent

Disjoint events do not happen at the same time

So if A occurs can B occur

Disjoint events are dependent

NO independent events cannot be disjoint

P(A or B) = P(A) + P(B) ndash P(A amp B)

How can you find the

probability of A amp B

P(A or B) = 45 + 35 - 45(35) = 06425

If independent

multiply

In a class there are 12 boys made up of 8 Seniors and 4 Juniors There are also 8 girls made up of 3 Seniors and 5 Juniors Find the probability of choosing a boy or a Senior

Note that choosing a boy and choosing a Senior are not disjoint (they can occur simultaneously)

P(boy or a senior) = P(Boy) + P(Senior) ndash P(Senior boy) = 12 11 8 15 3

20 20 20 20 4

Boy Girl Total

Senior 8 3 11

Junior 4 5 9

Total 12 8 20

Rule 6 At least one

The probability that at least one outcome happens is 1 minus the probability that no outcomes happen

P(at least 1) = 1 ndash P(none)

For a sales promotion the manufacturer places winning symbols under the caps of 10 of all Dr Pepper bottles You buy a six-pack What is the probability that you win something

P(at least one winning symbol) =

1 ndash P(no winning symbols) 1 - 96 = 4686

Suppose that 40 of cars in Fort Smith are manufactured in the United States 30 in Japan 10 in Germany and 20 in other countries

If cars are selected at random what is the probability that it is not US made

P(not US made) = 1 ndash P(US made) = 1 - 4 = 6

Suppose that 40 of cars in Fort Smith are manufactured in the United States 30 in Japan 10 in Germany and 20 in other countries

If cars are selected at random what is the probability that it is made in Japan or Germany

P(Japanese or German) = P(Japanese) + P(German) = 3 + 1 = 4

Suppose that 40 of cars in Fort Smith are manufactured in the United States 30 in Japan 10 in Germany and 20 in other countries

If cars are selected at random what is the probability that you see two in a row from Japan

P(2 Japanese in a row) = P(Japanese) and P(Japanese) = P(J) x P(J) = 3 x 3 = 09

Suppose that 40 of cars in Fort Smith are manufactured in the United States 30 in Japan 10 in Germany and 20 in other countries

If cars are selected at random what is the probability that none of three cars came from Germany

P(no Germany in three) = P(not G) x P(not G) x P(not G) = 9 x 9 x 9 = 729

Suppose that 40 of cars in Fort Smith are manufactured in the United States 30 in Japan 10 in Germany and 20 in other countries

If cars are selected at random what is the probability that at least one of three cars is US made

P(at least one US in three) = 1 ndash P(no US in three) = 1 ndash (6)(6)(6) = 784

Suppose that 40 of cars in Fort Smith are manufactured in the United States 30 in Japan 10 in Germany and 20 in other countries

If cars are selected at random what is the probability that the first Japanese car is the fourth one you choose

P(first J is the fourth car) = P(not J) x P(not J) x P(not J) x P(J) = (7)3 (3) = 1029

Watch out for

bull probabilities that donrsquot add up to 1

bull donrsquot add probabilities of events if they are not disjoint

bull donrsquot multiply probabilities of events if they are not independent

bull donrsquot confuse disjoint and independent

Rule 7 Conditional Probability

bull A probability that takes into account a given condition

P(A)

B)P(AA)|P(B

P(given)

P(and)A)|P(B

In a class there are 12 boys made up of 8 Seniors and 4 Juniors There are also 8 girls made up of 3 Seniors and 5 Juniors Find the probability of choosing a boy given that he is a Senior

Boy Girl Total

Senior 8 3 11

Junior 4 5 9

Total 12 8 20

P(Boy|Senior) = 811

P(Senior|Boy) = 812 = 23

Probabilities from two way tables

Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359

12) What is the probability that the driver is a student

359195

)( StudentP

Probabilities from two way tables

Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359

13) What is the probability that the driver drives a European car

35945

)( EuropeanP

Probabilities from two way tables

Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359

14) What is the probability that the driver drives an American or Asian car

Disjoint359102212

)(

AsianorAmericanP

Probabilities from two way tables

Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359

15) What is the probability that the driver is staff or drives an Asian car

Disjoint35947102164

)(

AsianorStaffP

Probabilities from two way tables

Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359

16) What is the probability that the driver is staff and drives an Asian car

35947

)( AsianandStaffP

Probabilities from two way tables

Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359

17) If the driver is a student what is the probability that they drive an American car

Condition195107

)|( StudentAmericanP

Probabilities from two way tables

Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359

18) What is the probability that the driver is a student if the driver drives a European car

Condition 4533

)|( EuropeanStudentP

Definition of Independent Events

Two events E and F are independent if and only if

P(F | E) = P(F) or P(E | F) = P(E)

EXAMPLE Illustrating Independent Events

The probability a randomly selected murder victim is male is 07515 The probability a randomly selected murder victim is male given that they are less than 18 years old is 06751

Since P(male) = 07515 and

P(male | lt 18 years old) = 06751

the events ldquomalerdquo and ldquoless than 18 years oldrdquo are not independent In fact knowing the victim is less than 18 years old decreases the probability that the victim is male

I draw one card and look

at it I tell you it is red What is the

probability it is a heart

P( heart | red) =

13P(heart and red) 152

26P(red) 252

Are ldquored cardrdquo and ldquospaderdquo mutually

exclusive Are they

independentA red card canrsquot be a spade so they ARE mutually exclusive

Are ldquored cardrdquo and

ldquoacerdquo mutually

exclusive Are they

independent2 aces are

red cards so they are

NOT mutually exclusive

Are ldquoface cardrdquo and

ldquokingrdquo mutually

exclusive Are they

independentKings are

Face cards so they are

NOT mutually exclusive

ldquoSlump I ainrsquot in no slump I just ainrsquot hittinrdquo

Yogi Berra

• PowerPoint Presentation
• Probability
• Law of Large Numbers
• Basic Rules of Probability
• Slide 5
• Slide 6
• Independent
• Slide 8
• Slide 9
• Slide 10
• Slide 11
• Slide 12
• Slide 13
• Slide 14
• Slide 15
• Slide 16
• Slide 17
• Slide 18
• Slide 19
• Slide 20
• Slide 21
• Slide 22
• Slide 23
• Slide 24
• Slide 25
• Slide 26
• Slide 27
• Slide 28
• Slide 29
• Slide 30
• Slide 31
• Slide 32
• Rule 7 Conditional Probability
• Slide 34
• Probabilities from two way tables
• Slide 36
• Slide 37
• Slide 38
• Slide 39
• Slide 40
• Slide 41
• Slide 42
• Slide 43
• Slide 44
• Slide 45
• Slide 46
• Slide 47
• Slide 48

)( FEP

)( FEP Mutually exclusive

)()( FPEP

Yes No

)()()( FEPFPEP

Given a deck of cards one

card is drawn What

is the probability that it is an ace or a red

cardP(ace or red) = P(ace) + P(red) ndash P(ace and red) =

52

4

52

26

52

2

52

28+ - =

)( FEP

)( FEP Mutually exclusive

)()( FPEP

Yes No

)()()( FEPFPEP

Independent Yes )()( FPEP

Ex 5)

If P(A) = 045 P(B) = 035 and A amp B are independent find P(A or B)Is A amp B disjoint

If A amp B are disjoint are they independent

Disjoint events do not happen at the same time

So if A occurs can B occur

Disjoint events are dependent

NO independent events cannot be disjoint

P(A or B) = P(A) + P(B) ndash P(A amp B)

How can you find the

probability of A amp B

P(A or B) = 45 + 35 - 45(35) = 06425

If independent

multiply

In a class there are 12 boys made up of 8 Seniors and 4 Juniors There are also 8 girls made up of 3 Seniors and 5 Juniors Find the probability of choosing a boy or a Senior

Note that choosing a boy and choosing a Senior are not disjoint (they can occur simultaneously)

P(boy or a senior) = P(Boy) + P(Senior) ndash P(Senior boy) = 12 11 8 15 3

20 20 20 20 4

Boy Girl Total

Senior 8 3 11

Junior 4 5 9

Total 12 8 20

Rule 6 At least one

The probability that at least one outcome happens is 1 minus the probability that no outcomes happen

P(at least 1) = 1 ndash P(none)

For a sales promotion the manufacturer places winning symbols under the caps of 10 of all Dr Pepper bottles You buy a six-pack What is the probability that you win something

P(at least one winning symbol) =

1 ndash P(no winning symbols) 1 - 96 = 4686

Suppose that 40 of cars in Fort Smith are manufactured in the United States 30 in Japan 10 in Germany and 20 in other countries

If cars are selected at random what is the probability that it is not US made

P(not US made) = 1 ndash P(US made) = 1 - 4 = 6

Suppose that 40 of cars in Fort Smith are manufactured in the United States 30 in Japan 10 in Germany and 20 in other countries

If cars are selected at random what is the probability that it is made in Japan or Germany

P(Japanese or German) = P(Japanese) + P(German) = 3 + 1 = 4

Suppose that 40 of cars in Fort Smith are manufactured in the United States 30 in Japan 10 in Germany and 20 in other countries

If cars are selected at random what is the probability that you see two in a row from Japan

P(2 Japanese in a row) = P(Japanese) and P(Japanese) = P(J) x P(J) = 3 x 3 = 09

Suppose that 40 of cars in Fort Smith are manufactured in the United States 30 in Japan 10 in Germany and 20 in other countries

If cars are selected at random what is the probability that none of three cars came from Germany

P(no Germany in three) = P(not G) x P(not G) x P(not G) = 9 x 9 x 9 = 729

Suppose that 40 of cars in Fort Smith are manufactured in the United States 30 in Japan 10 in Germany and 20 in other countries

If cars are selected at random what is the probability that at least one of three cars is US made

P(at least one US in three) = 1 ndash P(no US in three) = 1 ndash (6)(6)(6) = 784

Suppose that 40 of cars in Fort Smith are manufactured in the United States 30 in Japan 10 in Germany and 20 in other countries

If cars are selected at random what is the probability that the first Japanese car is the fourth one you choose

P(first J is the fourth car) = P(not J) x P(not J) x P(not J) x P(J) = (7)3 (3) = 1029

Watch out for

bull probabilities that donrsquot add up to 1

bull donrsquot add probabilities of events if they are not disjoint

bull donrsquot multiply probabilities of events if they are not independent

bull donrsquot confuse disjoint and independent

Rule 7 Conditional Probability

bull A probability that takes into account a given condition

P(A)

B)P(AA)|P(B

P(given)

P(and)A)|P(B

In a class there are 12 boys made up of 8 Seniors and 4 Juniors There are also 8 girls made up of 3 Seniors and 5 Juniors Find the probability of choosing a boy given that he is a Senior

Boy Girl Total

Senior 8 3 11

Junior 4 5 9

Total 12 8 20

P(Boy|Senior) = 811

P(Senior|Boy) = 812 = 23

Probabilities from two way tables

Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359

12) What is the probability that the driver is a student

359195

)( StudentP

Probabilities from two way tables

Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359

13) What is the probability that the driver drives a European car

35945

)( EuropeanP

Probabilities from two way tables

Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359

14) What is the probability that the driver drives an American or Asian car

Disjoint359102212

)(

AsianorAmericanP

Probabilities from two way tables

Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359

15) What is the probability that the driver is staff or drives an Asian car

Disjoint35947102164

)(

AsianorStaffP

Probabilities from two way tables

Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359

16) What is the probability that the driver is staff and drives an Asian car

35947

)( AsianandStaffP

Probabilities from two way tables

Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359

17) If the driver is a student what is the probability that they drive an American car

Condition195107

)|( StudentAmericanP

Probabilities from two way tables

Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359

18) What is the probability that the driver is a student if the driver drives a European car

Condition 4533

)|( EuropeanStudentP

Definition of Independent Events

Two events E and F are independent if and only if

P(F | E) = P(F) or P(E | F) = P(E)

EXAMPLE Illustrating Independent Events

The probability a randomly selected murder victim is male is 07515 The probability a randomly selected murder victim is male given that they are less than 18 years old is 06751

Since P(male) = 07515 and

P(male | lt 18 years old) = 06751

the events ldquomalerdquo and ldquoless than 18 years oldrdquo are not independent In fact knowing the victim is less than 18 years old decreases the probability that the victim is male

I draw one card and look

at it I tell you it is red What is the

probability it is a heart

P( heart | red) =

13P(heart and red) 152

26P(red) 252

Are ldquored cardrdquo and ldquospaderdquo mutually

exclusive Are they

independentA red card canrsquot be a spade so they ARE mutually exclusive

Are ldquored cardrdquo and

ldquoacerdquo mutually

exclusive Are they

independent2 aces are

red cards so they are

NOT mutually exclusive

Are ldquoface cardrdquo and

ldquokingrdquo mutually

exclusive Are they

independentKings are

Face cards so they are

NOT mutually exclusive

ldquoSlump I ainrsquot in no slump I just ainrsquot hittinrdquo

Yogi Berra

• PowerPoint Presentation
• Probability
• Law of Large Numbers
• Basic Rules of Probability
• Slide 5
• Slide 6
• Independent
• Slide 8
• Slide 9
• Slide 10
• Slide 11
• Slide 12
• Slide 13
• Slide 14
• Slide 15
• Slide 16
• Slide 17
• Slide 18
• Slide 19
• Slide 20
• Slide 21
• Slide 22
• Slide 23
• Slide 24
• Slide 25
• Slide 26
• Slide 27
• Slide 28
• Slide 29
• Slide 30
• Slide 31
• Slide 32
• Rule 7 Conditional Probability
• Slide 34
• Probabilities from two way tables
• Slide 36
• Slide 37
• Slide 38
• Slide 39
• Slide 40
• Slide 41
• Slide 42
• Slide 43
• Slide 44
• Slide 45
• Slide 46
• Slide 47
• Slide 48

Given a deck of cards one

card is drawn What

is the probability that it is an ace or a red

cardP(ace or red) = P(ace) + P(red) ndash P(ace and red) =

52

4

52

26

52

2

52

28+ - =

)( FEP

)( FEP Mutually exclusive

)()( FPEP

Yes No

)()()( FEPFPEP

Independent Yes )()( FPEP

Ex 5)

If P(A) = 045 P(B) = 035 and A amp B are independent find P(A or B)Is A amp B disjoint

If A amp B are disjoint are they independent

Disjoint events do not happen at the same time

So if A occurs can B occur

Disjoint events are dependent

NO independent events cannot be disjoint

P(A or B) = P(A) + P(B) ndash P(A amp B)

How can you find the

probability of A amp B

P(A or B) = 45 + 35 - 45(35) = 06425

If independent

multiply

In a class there are 12 boys made up of 8 Seniors and 4 Juniors There are also 8 girls made up of 3 Seniors and 5 Juniors Find the probability of choosing a boy or a Senior

Note that choosing a boy and choosing a Senior are not disjoint (they can occur simultaneously)

P(boy or a senior) = P(Boy) + P(Senior) ndash P(Senior boy) = 12 11 8 15 3

20 20 20 20 4

Boy Girl Total

Senior 8 3 11

Junior 4 5 9

Total 12 8 20

Rule 6 At least one

The probability that at least one outcome happens is 1 minus the probability that no outcomes happen

P(at least 1) = 1 ndash P(none)

For a sales promotion the manufacturer places winning symbols under the caps of 10 of all Dr Pepper bottles You buy a six-pack What is the probability that you win something

P(at least one winning symbol) =

1 ndash P(no winning symbols) 1 - 96 = 4686

Suppose that 40 of cars in Fort Smith are manufactured in the United States 30 in Japan 10 in Germany and 20 in other countries

If cars are selected at random what is the probability that it is not US made

P(not US made) = 1 ndash P(US made) = 1 - 4 = 6

Suppose that 40 of cars in Fort Smith are manufactured in the United States 30 in Japan 10 in Germany and 20 in other countries

If cars are selected at random what is the probability that it is made in Japan or Germany

P(Japanese or German) = P(Japanese) + P(German) = 3 + 1 = 4

Suppose that 40 of cars in Fort Smith are manufactured in the United States 30 in Japan 10 in Germany and 20 in other countries

If cars are selected at random what is the probability that you see two in a row from Japan

P(2 Japanese in a row) = P(Japanese) and P(Japanese) = P(J) x P(J) = 3 x 3 = 09

Suppose that 40 of cars in Fort Smith are manufactured in the United States 30 in Japan 10 in Germany and 20 in other countries

If cars are selected at random what is the probability that none of three cars came from Germany

P(no Germany in three) = P(not G) x P(not G) x P(not G) = 9 x 9 x 9 = 729

Suppose that 40 of cars in Fort Smith are manufactured in the United States 30 in Japan 10 in Germany and 20 in other countries

If cars are selected at random what is the probability that at least one of three cars is US made

P(at least one US in three) = 1 ndash P(no US in three) = 1 ndash (6)(6)(6) = 784

Suppose that 40 of cars in Fort Smith are manufactured in the United States 30 in Japan 10 in Germany and 20 in other countries

If cars are selected at random what is the probability that the first Japanese car is the fourth one you choose

P(first J is the fourth car) = P(not J) x P(not J) x P(not J) x P(J) = (7)3 (3) = 1029

Watch out for

bull probabilities that donrsquot add up to 1

bull donrsquot add probabilities of events if they are not disjoint

bull donrsquot multiply probabilities of events if they are not independent

bull donrsquot confuse disjoint and independent

Rule 7 Conditional Probability

bull A probability that takes into account a given condition

P(A)

B)P(AA)|P(B

P(given)

P(and)A)|P(B

In a class there are 12 boys made up of 8 Seniors and 4 Juniors There are also 8 girls made up of 3 Seniors and 5 Juniors Find the probability of choosing a boy given that he is a Senior

Boy Girl Total

Senior 8 3 11

Junior 4 5 9

Total 12 8 20

P(Boy|Senior) = 811

P(Senior|Boy) = 812 = 23

Probabilities from two way tables

Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359

12) What is the probability that the driver is a student

359195

)( StudentP

Probabilities from two way tables

Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359

13) What is the probability that the driver drives a European car

35945

)( EuropeanP

Probabilities from two way tables

Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359

14) What is the probability that the driver drives an American or Asian car

Disjoint359102212

)(

AsianorAmericanP

Probabilities from two way tables

Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359

15) What is the probability that the driver is staff or drives an Asian car

Disjoint35947102164

)(

AsianorStaffP

Probabilities from two way tables

Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359

16) What is the probability that the driver is staff and drives an Asian car

35947

)( AsianandStaffP

Probabilities from two way tables

Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359

17) If the driver is a student what is the probability that they drive an American car

Condition195107

)|( StudentAmericanP

Probabilities from two way tables

Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359

18) What is the probability that the driver is a student if the driver drives a European car

Condition 4533

)|( EuropeanStudentP

Definition of Independent Events

Two events E and F are independent if and only if

P(F | E) = P(F) or P(E | F) = P(E)

EXAMPLE Illustrating Independent Events

The probability a randomly selected murder victim is male is 07515 The probability a randomly selected murder victim is male given that they are less than 18 years old is 06751

Since P(male) = 07515 and

P(male | lt 18 years old) = 06751

the events ldquomalerdquo and ldquoless than 18 years oldrdquo are not independent In fact knowing the victim is less than 18 years old decreases the probability that the victim is male

I draw one card and look

at it I tell you it is red What is the

probability it is a heart

P( heart | red) =

13P(heart and red) 152

26P(red) 252

Are ldquored cardrdquo and ldquospaderdquo mutually

exclusive Are they

independentA red card canrsquot be a spade so they ARE mutually exclusive

Are ldquored cardrdquo and

ldquoacerdquo mutually

exclusive Are they

independent2 aces are

red cards so they are

NOT mutually exclusive

Are ldquoface cardrdquo and

ldquokingrdquo mutually

exclusive Are they

independentKings are

Face cards so they are

NOT mutually exclusive

ldquoSlump I ainrsquot in no slump I just ainrsquot hittinrdquo

Yogi Berra

• PowerPoint Presentation
• Probability
• Law of Large Numbers
• Basic Rules of Probability
• Slide 5
• Slide 6
• Independent
• Slide 8
• Slide 9
• Slide 10
• Slide 11
• Slide 12
• Slide 13
• Slide 14
• Slide 15
• Slide 16
• Slide 17
• Slide 18
• Slide 19
• Slide 20
• Slide 21
• Slide 22
• Slide 23
• Slide 24
• Slide 25
• Slide 26
• Slide 27
• Slide 28
• Slide 29
• Slide 30
• Slide 31
• Slide 32
• Rule 7 Conditional Probability
• Slide 34
• Probabilities from two way tables
• Slide 36
• Slide 37
• Slide 38
• Slide 39
• Slide 40
• Slide 41
• Slide 42
• Slide 43
• Slide 44
• Slide 45
• Slide 46
• Slide 47
• Slide 48

)( FEP

)( FEP Mutually exclusive

)()( FPEP

Yes No

)()()( FEPFPEP

Independent Yes )()( FPEP

Ex 5)

If P(A) = 045 P(B) = 035 and A amp B are independent find P(A or B)Is A amp B disjoint

If A amp B are disjoint are they independent

Disjoint events do not happen at the same time

So if A occurs can B occur

Disjoint events are dependent

NO independent events cannot be disjoint

P(A or B) = P(A) + P(B) ndash P(A amp B)

How can you find the

probability of A amp B

P(A or B) = 45 + 35 - 45(35) = 06425

If independent

multiply

In a class there are 12 boys made up of 8 Seniors and 4 Juniors There are also 8 girls made up of 3 Seniors and 5 Juniors Find the probability of choosing a boy or a Senior

Note that choosing a boy and choosing a Senior are not disjoint (they can occur simultaneously)

P(boy or a senior) = P(Boy) + P(Senior) ndash P(Senior boy) = 12 11 8 15 3

20 20 20 20 4

Boy Girl Total

Senior 8 3 11

Junior 4 5 9

Total 12 8 20

Rule 6 At least one

The probability that at least one outcome happens is 1 minus the probability that no outcomes happen

P(at least 1) = 1 ndash P(none)

For a sales promotion the manufacturer places winning symbols under the caps of 10 of all Dr Pepper bottles You buy a six-pack What is the probability that you win something

P(at least one winning symbol) =

1 ndash P(no winning symbols) 1 - 96 = 4686

Suppose that 40 of cars in Fort Smith are manufactured in the United States 30 in Japan 10 in Germany and 20 in other countries

If cars are selected at random what is the probability that it is not US made

P(not US made) = 1 ndash P(US made) = 1 - 4 = 6

Suppose that 40 of cars in Fort Smith are manufactured in the United States 30 in Japan 10 in Germany and 20 in other countries

If cars are selected at random what is the probability that it is made in Japan or Germany

P(Japanese or German) = P(Japanese) + P(German) = 3 + 1 = 4

Suppose that 40 of cars in Fort Smith are manufactured in the United States 30 in Japan 10 in Germany and 20 in other countries

If cars are selected at random what is the probability that you see two in a row from Japan

P(2 Japanese in a row) = P(Japanese) and P(Japanese) = P(J) x P(J) = 3 x 3 = 09

Suppose that 40 of cars in Fort Smith are manufactured in the United States 30 in Japan 10 in Germany and 20 in other countries

If cars are selected at random what is the probability that none of three cars came from Germany

P(no Germany in three) = P(not G) x P(not G) x P(not G) = 9 x 9 x 9 = 729

Suppose that 40 of cars in Fort Smith are manufactured in the United States 30 in Japan 10 in Germany and 20 in other countries

If cars are selected at random what is the probability that at least one of three cars is US made

P(at least one US in three) = 1 ndash P(no US in three) = 1 ndash (6)(6)(6) = 784

Suppose that 40 of cars in Fort Smith are manufactured in the United States 30 in Japan 10 in Germany and 20 in other countries

If cars are selected at random what is the probability that the first Japanese car is the fourth one you choose

P(first J is the fourth car) = P(not J) x P(not J) x P(not J) x P(J) = (7)3 (3) = 1029

Watch out for

bull probabilities that donrsquot add up to 1

bull donrsquot add probabilities of events if they are not disjoint

bull donrsquot multiply probabilities of events if they are not independent

bull donrsquot confuse disjoint and independent

Rule 7 Conditional Probability

bull A probability that takes into account a given condition

P(A)

B)P(AA)|P(B

P(given)

P(and)A)|P(B

In a class there are 12 boys made up of 8 Seniors and 4 Juniors There are also 8 girls made up of 3 Seniors and 5 Juniors Find the probability of choosing a boy given that he is a Senior

Boy Girl Total

Senior 8 3 11

Junior 4 5 9

Total 12 8 20

P(Boy|Senior) = 811

P(Senior|Boy) = 812 = 23

Probabilities from two way tables

Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359

12) What is the probability that the driver is a student

359195

)( StudentP

Probabilities from two way tables

Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359

13) What is the probability that the driver drives a European car

35945

)( EuropeanP

Probabilities from two way tables

Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359

14) What is the probability that the driver drives an American or Asian car

Disjoint359102212

)(

AsianorAmericanP

Probabilities from two way tables

Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359

15) What is the probability that the driver is staff or drives an Asian car

Disjoint35947102164

)(

AsianorStaffP

Probabilities from two way tables

Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359

16) What is the probability that the driver is staff and drives an Asian car

35947

)( AsianandStaffP

Probabilities from two way tables

Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359

17) If the driver is a student what is the probability that they drive an American car

Condition195107

)|( StudentAmericanP

Probabilities from two way tables

Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359

18) What is the probability that the driver is a student if the driver drives a European car

Condition 4533

)|( EuropeanStudentP

Definition of Independent Events

Two events E and F are independent if and only if

P(F | E) = P(F) or P(E | F) = P(E)

EXAMPLE Illustrating Independent Events

The probability a randomly selected murder victim is male is 07515 The probability a randomly selected murder victim is male given that they are less than 18 years old is 06751

Since P(male) = 07515 and

P(male | lt 18 years old) = 06751

the events ldquomalerdquo and ldquoless than 18 years oldrdquo are not independent In fact knowing the victim is less than 18 years old decreases the probability that the victim is male

I draw one card and look

at it I tell you it is red What is the

probability it is a heart

P( heart | red) =

13P(heart and red) 152

26P(red) 252

Are ldquored cardrdquo and ldquospaderdquo mutually

exclusive Are they

independentA red card canrsquot be a spade so they ARE mutually exclusive

Are ldquored cardrdquo and

ldquoacerdquo mutually

exclusive Are they

independent2 aces are

red cards so they are

NOT mutually exclusive

Are ldquoface cardrdquo and

ldquokingrdquo mutually

exclusive Are they

independentKings are

Face cards so they are

NOT mutually exclusive

ldquoSlump I ainrsquot in no slump I just ainrsquot hittinrdquo

Yogi Berra

• PowerPoint Presentation
• Probability
• Law of Large Numbers
• Basic Rules of Probability
• Slide 5
• Slide 6
• Independent
• Slide 8
• Slide 9
• Slide 10
• Slide 11
• Slide 12
• Slide 13
• Slide 14
• Slide 15
• Slide 16
• Slide 17
• Slide 18
• Slide 19
• Slide 20
• Slide 21
• Slide 22
• Slide 23
• Slide 24
• Slide 25
• Slide 26
• Slide 27
• Slide 28
• Slide 29
• Slide 30
• Slide 31
• Slide 32
• Rule 7 Conditional Probability
• Slide 34
• Probabilities from two way tables
• Slide 36
• Slide 37
• Slide 38
• Slide 39
• Slide 40
• Slide 41
• Slide 42
• Slide 43
• Slide 44
• Slide 45
• Slide 46
• Slide 47
• Slide 48

Ex 5)

If P(A) = 045 P(B) = 035 and A amp B are independent find P(A or B)Is A amp B disjoint

If A amp B are disjoint are they independent

Disjoint events do not happen at the same time

So if A occurs can B occur

Disjoint events are dependent

NO independent events cannot be disjoint

P(A or B) = P(A) + P(B) ndash P(A amp B)

How can you find the

probability of A amp B

P(A or B) = 45 + 35 - 45(35) = 06425

If independent

multiply

In a class there are 12 boys made up of 8 Seniors and 4 Juniors There are also 8 girls made up of 3 Seniors and 5 Juniors Find the probability of choosing a boy or a Senior

Note that choosing a boy and choosing a Senior are not disjoint (they can occur simultaneously)

P(boy or a senior) = P(Boy) + P(Senior) ndash P(Senior boy) = 12 11 8 15 3

20 20 20 20 4

Boy Girl Total

Senior 8 3 11

Junior 4 5 9

Total 12 8 20

Rule 6 At least one

The probability that at least one outcome happens is 1 minus the probability that no outcomes happen

P(at least 1) = 1 ndash P(none)

For a sales promotion the manufacturer places winning symbols under the caps of 10 of all Dr Pepper bottles You buy a six-pack What is the probability that you win something

P(at least one winning symbol) =

1 ndash P(no winning symbols) 1 - 96 = 4686

Suppose that 40 of cars in Fort Smith are manufactured in the United States 30 in Japan 10 in Germany and 20 in other countries

If cars are selected at random what is the probability that it is not US made

P(not US made) = 1 ndash P(US made) = 1 - 4 = 6

Suppose that 40 of cars in Fort Smith are manufactured in the United States 30 in Japan 10 in Germany and 20 in other countries

If cars are selected at random what is the probability that it is made in Japan or Germany

P(Japanese or German) = P(Japanese) + P(German) = 3 + 1 = 4

Suppose that 40 of cars in Fort Smith are manufactured in the United States 30 in Japan 10 in Germany and 20 in other countries

If cars are selected at random what is the probability that you see two in a row from Japan

P(2 Japanese in a row) = P(Japanese) and P(Japanese) = P(J) x P(J) = 3 x 3 = 09

Suppose that 40 of cars in Fort Smith are manufactured in the United States 30 in Japan 10 in Germany and 20 in other countries

If cars are selected at random what is the probability that none of three cars came from Germany

P(no Germany in three) = P(not G) x P(not G) x P(not G) = 9 x 9 x 9 = 729

Suppose that 40 of cars in Fort Smith are manufactured in the United States 30 in Japan 10 in Germany and 20 in other countries

If cars are selected at random what is the probability that at least one of three cars is US made

P(at least one US in three) = 1 ndash P(no US in three) = 1 ndash (6)(6)(6) = 784

Suppose that 40 of cars in Fort Smith are manufactured in the United States 30 in Japan 10 in Germany and 20 in other countries

If cars are selected at random what is the probability that the first Japanese car is the fourth one you choose

P(first J is the fourth car) = P(not J) x P(not J) x P(not J) x P(J) = (7)3 (3) = 1029

Watch out for

bull probabilities that donrsquot add up to 1

bull donrsquot add probabilities of events if they are not disjoint

bull donrsquot multiply probabilities of events if they are not independent

bull donrsquot confuse disjoint and independent

Rule 7 Conditional Probability

bull A probability that takes into account a given condition

P(A)

B)P(AA)|P(B

P(given)

P(and)A)|P(B

In a class there are 12 boys made up of 8 Seniors and 4 Juniors There are also 8 girls made up of 3 Seniors and 5 Juniors Find the probability of choosing a boy given that he is a Senior

Boy Girl Total

Senior 8 3 11

Junior 4 5 9

Total 12 8 20

P(Boy|Senior) = 811

P(Senior|Boy) = 812 = 23

Probabilities from two way tables

Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359

12) What is the probability that the driver is a student

359195

)( StudentP

Probabilities from two way tables

Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359

13) What is the probability that the driver drives a European car

35945

)( EuropeanP

Probabilities from two way tables

Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359

14) What is the probability that the driver drives an American or Asian car

Disjoint359102212

)(

AsianorAmericanP

Probabilities from two way tables

Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359

15) What is the probability that the driver is staff or drives an Asian car

Disjoint35947102164

)(

AsianorStaffP

Probabilities from two way tables

Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359

16) What is the probability that the driver is staff and drives an Asian car

35947

)( AsianandStaffP

Probabilities from two way tables

Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359

17) If the driver is a student what is the probability that they drive an American car

Condition195107

)|( StudentAmericanP

Probabilities from two way tables

Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359

18) What is the probability that the driver is a student if the driver drives a European car

Condition 4533

)|( EuropeanStudentP

Definition of Independent Events

Two events E and F are independent if and only if

P(F | E) = P(F) or P(E | F) = P(E)

EXAMPLE Illustrating Independent Events

The probability a randomly selected murder victim is male is 07515 The probability a randomly selected murder victim is male given that they are less than 18 years old is 06751

Since P(male) = 07515 and

P(male | lt 18 years old) = 06751

the events ldquomalerdquo and ldquoless than 18 years oldrdquo are not independent In fact knowing the victim is less than 18 years old decreases the probability that the victim is male

I draw one card and look

at it I tell you it is red What is the

probability it is a heart

P( heart | red) =

13P(heart and red) 152

26P(red) 252

Are ldquored cardrdquo and ldquospaderdquo mutually

exclusive Are they

independentA red card canrsquot be a spade so they ARE mutually exclusive

Are ldquored cardrdquo and

ldquoacerdquo mutually

exclusive Are they

independent2 aces are

red cards so they are

NOT mutually exclusive

Are ldquoface cardrdquo and

ldquokingrdquo mutually

exclusive Are they

independentKings are

Face cards so they are

NOT mutually exclusive

ldquoSlump I ainrsquot in no slump I just ainrsquot hittinrdquo

Yogi Berra

• PowerPoint Presentation
• Probability
• Law of Large Numbers
• Basic Rules of Probability
• Slide 5
• Slide 6
• Independent
• Slide 8
• Slide 9
• Slide 10
• Slide 11
• Slide 12
• Slide 13
• Slide 14
• Slide 15
• Slide 16
• Slide 17
• Slide 18
• Slide 19
• Slide 20
• Slide 21
• Slide 22
• Slide 23
• Slide 24
• Slide 25
• Slide 26
• Slide 27
• Slide 28
• Slide 29
• Slide 30
• Slide 31
• Slide 32
• Rule 7 Conditional Probability
• Slide 34
• Probabilities from two way tables
• Slide 36
• Slide 37
• Slide 38
• Slide 39
• Slide 40
• Slide 41
• Slide 42
• Slide 43
• Slide 44
• Slide 45
• Slide 46
• Slide 47
• Slide 48

In a class there are 12 boys made up of 8 Seniors and 4 Juniors There are also 8 girls made up of 3 Seniors and 5 Juniors Find the probability of choosing a boy or a Senior

Note that choosing a boy and choosing a Senior are not disjoint (they can occur simultaneously)

P(boy or a senior) = P(Boy) + P(Senior) ndash P(Senior boy) = 12 11 8 15 3

20 20 20 20 4

Boy Girl Total

Senior 8 3 11

Junior 4 5 9

Total 12 8 20

Rule 6 At least one

The probability that at least one outcome happens is 1 minus the probability that no outcomes happen

P(at least 1) = 1 ndash P(none)

For a sales promotion the manufacturer places winning symbols under the caps of 10 of all Dr Pepper bottles You buy a six-pack What is the probability that you win something

P(at least one winning symbol) =

1 ndash P(no winning symbols) 1 - 96 = 4686

Suppose that 40 of cars in Fort Smith are manufactured in the United States 30 in Japan 10 in Germany and 20 in other countries

If cars are selected at random what is the probability that it is not US made

P(not US made) = 1 ndash P(US made) = 1 - 4 = 6

Suppose that 40 of cars in Fort Smith are manufactured in the United States 30 in Japan 10 in Germany and 20 in other countries

If cars are selected at random what is the probability that it is made in Japan or Germany

P(Japanese or German) = P(Japanese) + P(German) = 3 + 1 = 4

Suppose that 40 of cars in Fort Smith are manufactured in the United States 30 in Japan 10 in Germany and 20 in other countries

If cars are selected at random what is the probability that you see two in a row from Japan

P(2 Japanese in a row) = P(Japanese) and P(Japanese) = P(J) x P(J) = 3 x 3 = 09

Suppose that 40 of cars in Fort Smith are manufactured in the United States 30 in Japan 10 in Germany and 20 in other countries

If cars are selected at random what is the probability that none of three cars came from Germany

P(no Germany in three) = P(not G) x P(not G) x P(not G) = 9 x 9 x 9 = 729

Suppose that 40 of cars in Fort Smith are manufactured in the United States 30 in Japan 10 in Germany and 20 in other countries

If cars are selected at random what is the probability that at least one of three cars is US made

P(at least one US in three) = 1 ndash P(no US in three) = 1 ndash (6)(6)(6) = 784

Suppose that 40 of cars in Fort Smith are manufactured in the United States 30 in Japan 10 in Germany and 20 in other countries

If cars are selected at random what is the probability that the first Japanese car is the fourth one you choose

P(first J is the fourth car) = P(not J) x P(not J) x P(not J) x P(J) = (7)3 (3) = 1029

Watch out for

bull probabilities that donrsquot add up to 1

bull donrsquot add probabilities of events if they are not disjoint

bull donrsquot multiply probabilities of events if they are not independent

bull donrsquot confuse disjoint and independent

Rule 7 Conditional Probability

bull A probability that takes into account a given condition

P(A)

B)P(AA)|P(B

P(given)

P(and)A)|P(B

In a class there are 12 boys made up of 8 Seniors and 4 Juniors There are also 8 girls made up of 3 Seniors and 5 Juniors Find the probability of choosing a boy given that he is a Senior

Boy Girl Total

Senior 8 3 11

Junior 4 5 9

Total 12 8 20

P(Boy|Senior) = 811

P(Senior|Boy) = 812 = 23

Probabilities from two way tables

Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359

12) What is the probability that the driver is a student

359195

)( StudentP

Probabilities from two way tables

Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359

13) What is the probability that the driver drives a European car

35945

)( EuropeanP

Probabilities from two way tables

Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359

14) What is the probability that the driver drives an American or Asian car

Disjoint359102212

)(

AsianorAmericanP

Probabilities from two way tables

Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359

15) What is the probability that the driver is staff or drives an Asian car

Disjoint35947102164

)(

AsianorStaffP

Probabilities from two way tables

Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359

16) What is the probability that the driver is staff and drives an Asian car

35947

)( AsianandStaffP

Probabilities from two way tables

Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359

17) If the driver is a student what is the probability that they drive an American car

Condition195107

)|( StudentAmericanP

Probabilities from two way tables

Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359

18) What is the probability that the driver is a student if the driver drives a European car

Condition 4533

)|( EuropeanStudentP

Definition of Independent Events

Two events E and F are independent if and only if

P(F | E) = P(F) or P(E | F) = P(E)

EXAMPLE Illustrating Independent Events

The probability a randomly selected murder victim is male is 07515 The probability a randomly selected murder victim is male given that they are less than 18 years old is 06751

Since P(male) = 07515 and

P(male | lt 18 years old) = 06751

the events ldquomalerdquo and ldquoless than 18 years oldrdquo are not independent In fact knowing the victim is less than 18 years old decreases the probability that the victim is male

I draw one card and look

at it I tell you it is red What is the

probability it is a heart

P( heart | red) =

13P(heart and red) 152

26P(red) 252

Are ldquored cardrdquo and ldquospaderdquo mutually

exclusive Are they

independentA red card canrsquot be a spade so they ARE mutually exclusive

Are ldquored cardrdquo and

ldquoacerdquo mutually

exclusive Are they

independent2 aces are

red cards so they are

NOT mutually exclusive

Are ldquoface cardrdquo and

ldquokingrdquo mutually

exclusive Are they

independentKings are

Face cards so they are

NOT mutually exclusive

ldquoSlump I ainrsquot in no slump I just ainrsquot hittinrdquo

Yogi Berra

• PowerPoint Presentation
• Probability
• Law of Large Numbers
• Basic Rules of Probability
• Slide 5
• Slide 6
• Independent
• Slide 8
• Slide 9
• Slide 10
• Slide 11
• Slide 12
• Slide 13
• Slide 14
• Slide 15
• Slide 16
• Slide 17
• Slide 18
• Slide 19
• Slide 20
• Slide 21
• Slide 22
• Slide 23
• Slide 24
• Slide 25
• Slide 26
• Slide 27
• Slide 28
• Slide 29
• Slide 30
• Slide 31
• Slide 32
• Rule 7 Conditional Probability
• Slide 34
• Probabilities from two way tables
• Slide 36
• Slide 37
• Slide 38
• Slide 39
• Slide 40
• Slide 41
• Slide 42
• Slide 43
• Slide 44
• Slide 45
• Slide 46
• Slide 47
• Slide 48

Rule 6 At least one

The probability that at least one outcome happens is 1 minus the probability that no outcomes happen

P(at least 1) = 1 ndash P(none)

For a sales promotion the manufacturer places winning symbols under the caps of 10 of all Dr Pepper bottles You buy a six-pack What is the probability that you win something

P(at least one winning symbol) =

1 ndash P(no winning symbols) 1 - 96 = 4686

Suppose that 40 of cars in Fort Smith are manufactured in the United States 30 in Japan 10 in Germany and 20 in other countries

If cars are selected at random what is the probability that it is not US made

P(not US made) = 1 ndash P(US made) = 1 - 4 = 6

Suppose that 40 of cars in Fort Smith are manufactured in the United States 30 in Japan 10 in Germany and 20 in other countries

If cars are selected at random what is the probability that it is made in Japan or Germany

P(Japanese or German) = P(Japanese) + P(German) = 3 + 1 = 4

Suppose that 40 of cars in Fort Smith are manufactured in the United States 30 in Japan 10 in Germany and 20 in other countries

If cars are selected at random what is the probability that you see two in a row from Japan

P(2 Japanese in a row) = P(Japanese) and P(Japanese) = P(J) x P(J) = 3 x 3 = 09

Suppose that 40 of cars in Fort Smith are manufactured in the United States 30 in Japan 10 in Germany and 20 in other countries

If cars are selected at random what is the probability that none of three cars came from Germany

P(no Germany in three) = P(not G) x P(not G) x P(not G) = 9 x 9 x 9 = 729

Suppose that 40 of cars in Fort Smith are manufactured in the United States 30 in Japan 10 in Germany and 20 in other countries

If cars are selected at random what is the probability that at least one of three cars is US made

P(at least one US in three) = 1 ndash P(no US in three) = 1 ndash (6)(6)(6) = 784

Suppose that 40 of cars in Fort Smith are manufactured in the United States 30 in Japan 10 in Germany and 20 in other countries

If cars are selected at random what is the probability that the first Japanese car is the fourth one you choose

P(first J is the fourth car) = P(not J) x P(not J) x P(not J) x P(J) = (7)3 (3) = 1029

Watch out for

bull probabilities that donrsquot add up to 1

bull donrsquot add probabilities of events if they are not disjoint

bull donrsquot multiply probabilities of events if they are not independent

bull donrsquot confuse disjoint and independent

Rule 7 Conditional Probability

bull A probability that takes into account a given condition

P(A)

B)P(AA)|P(B

P(given)

P(and)A)|P(B

In a class there are 12 boys made up of 8 Seniors and 4 Juniors There are also 8 girls made up of 3 Seniors and 5 Juniors Find the probability of choosing a boy given that he is a Senior

Boy Girl Total

Senior 8 3 11

Junior 4 5 9

Total 12 8 20

P(Boy|Senior) = 811

P(Senior|Boy) = 812 = 23

Probabilities from two way tables

Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359

12) What is the probability that the driver is a student

359195

)( StudentP

Probabilities from two way tables

Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359

13) What is the probability that the driver drives a European car

35945

)( EuropeanP

Probabilities from two way tables

Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359

14) What is the probability that the driver drives an American or Asian car

Disjoint359102212

)(

AsianorAmericanP

Probabilities from two way tables

Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359

15) What is the probability that the driver is staff or drives an Asian car

Disjoint35947102164

)(

AsianorStaffP

Probabilities from two way tables

Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359

16) What is the probability that the driver is staff and drives an Asian car

35947

)( AsianandStaffP

Probabilities from two way tables

Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359

17) If the driver is a student what is the probability that they drive an American car

Condition195107

)|( StudentAmericanP

Probabilities from two way tables

Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359

18) What is the probability that the driver is a student if the driver drives a European car

Condition 4533

)|( EuropeanStudentP

Definition of Independent Events

Two events E and F are independent if and only if

P(F | E) = P(F) or P(E | F) = P(E)

EXAMPLE Illustrating Independent Events

The probability a randomly selected murder victim is male is 07515 The probability a randomly selected murder victim is male given that they are less than 18 years old is 06751

Since P(male) = 07515 and

P(male | lt 18 years old) = 06751

the events ldquomalerdquo and ldquoless than 18 years oldrdquo are not independent In fact knowing the victim is less than 18 years old decreases the probability that the victim is male

I draw one card and look

at it I tell you it is red What is the

probability it is a heart

P( heart | red) =

13P(heart and red) 152

26P(red) 252

Are ldquored cardrdquo and ldquospaderdquo mutually

exclusive Are they

independentA red card canrsquot be a spade so they ARE mutually exclusive

Are ldquored cardrdquo and

ldquoacerdquo mutually

exclusive Are they

independent2 aces are

red cards so they are

NOT mutually exclusive

Are ldquoface cardrdquo and

ldquokingrdquo mutually

exclusive Are they

independentKings are

Face cards so they are

NOT mutually exclusive

ldquoSlump I ainrsquot in no slump I just ainrsquot hittinrdquo

Yogi Berra

• PowerPoint Presentation
• Probability
• Law of Large Numbers
• Basic Rules of Probability
• Slide 5
• Slide 6
• Independent
• Slide 8
• Slide 9
• Slide 10
• Slide 11
• Slide 12
• Slide 13
• Slide 14
• Slide 15
• Slide 16
• Slide 17
• Slide 18
• Slide 19
• Slide 20
• Slide 21
• Slide 22
• Slide 23
• Slide 24
• Slide 25
• Slide 26
• Slide 27
• Slide 28
• Slide 29
• Slide 30
• Slide 31
• Slide 32
• Rule 7 Conditional Probability
• Slide 34
• Probabilities from two way tables
• Slide 36
• Slide 37
• Slide 38
• Slide 39
• Slide 40
• Slide 41
• Slide 42
• Slide 43
• Slide 44
• Slide 45
• Slide 46
• Slide 47
• Slide 48

For a sales promotion the manufacturer places winning symbols under the caps of 10 of all Dr Pepper bottles You buy a six-pack What is the probability that you win something

P(at least one winning symbol) =

1 ndash P(no winning symbols) 1 - 96 = 4686

Suppose that 40 of cars in Fort Smith are manufactured in the United States 30 in Japan 10 in Germany and 20 in other countries

If cars are selected at random what is the probability that it is not US made

P(not US made) = 1 ndash P(US made) = 1 - 4 = 6

Suppose that 40 of cars in Fort Smith are manufactured in the United States 30 in Japan 10 in Germany and 20 in other countries

If cars are selected at random what is the probability that it is made in Japan or Germany

P(Japanese or German) = P(Japanese) + P(German) = 3 + 1 = 4

Suppose that 40 of cars in Fort Smith are manufactured in the United States 30 in Japan 10 in Germany and 20 in other countries

If cars are selected at random what is the probability that you see two in a row from Japan

P(2 Japanese in a row) = P(Japanese) and P(Japanese) = P(J) x P(J) = 3 x 3 = 09

Suppose that 40 of cars in Fort Smith are manufactured in the United States 30 in Japan 10 in Germany and 20 in other countries

If cars are selected at random what is the probability that none of three cars came from Germany

P(no Germany in three) = P(not G) x P(not G) x P(not G) = 9 x 9 x 9 = 729

Suppose that 40 of cars in Fort Smith are manufactured in the United States 30 in Japan 10 in Germany and 20 in other countries

If cars are selected at random what is the probability that at least one of three cars is US made

P(at least one US in three) = 1 ndash P(no US in three) = 1 ndash (6)(6)(6) = 784

Suppose that 40 of cars in Fort Smith are manufactured in the United States 30 in Japan 10 in Germany and 20 in other countries

If cars are selected at random what is the probability that the first Japanese car is the fourth one you choose

P(first J is the fourth car) = P(not J) x P(not J) x P(not J) x P(J) = (7)3 (3) = 1029

Watch out for

bull probabilities that donrsquot add up to 1

bull donrsquot add probabilities of events if they are not disjoint

bull donrsquot multiply probabilities of events if they are not independent

bull donrsquot confuse disjoint and independent

Rule 7 Conditional Probability

bull A probability that takes into account a given condition

P(A)

B)P(AA)|P(B

P(given)

P(and)A)|P(B

In a class there are 12 boys made up of 8 Seniors and 4 Juniors There are also 8 girls made up of 3 Seniors and 5 Juniors Find the probability of choosing a boy given that he is a Senior

Boy Girl Total

Senior 8 3 11

Junior 4 5 9

Total 12 8 20

P(Boy|Senior) = 811

P(Senior|Boy) = 812 = 23

Probabilities from two way tables

Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359

12) What is the probability that the driver is a student

359195

)( StudentP

Probabilities from two way tables

Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359

13) What is the probability that the driver drives a European car

35945

)( EuropeanP

Probabilities from two way tables

Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359

14) What is the probability that the driver drives an American or Asian car

Disjoint359102212

)(

AsianorAmericanP

Probabilities from two way tables

Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359

15) What is the probability that the driver is staff or drives an Asian car

Disjoint35947102164

)(

AsianorStaffP

Probabilities from two way tables

Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359

16) What is the probability that the driver is staff and drives an Asian car

35947

)( AsianandStaffP

Probabilities from two way tables

Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359

17) If the driver is a student what is the probability that they drive an American car

Condition195107

)|( StudentAmericanP

Probabilities from two way tables

Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359

18) What is the probability that the driver is a student if the driver drives a European car

Condition 4533

)|( EuropeanStudentP

Definition of Independent Events

Two events E and F are independent if and only if

P(F | E) = P(F) or P(E | F) = P(E)

EXAMPLE Illustrating Independent Events

The probability a randomly selected murder victim is male is 07515 The probability a randomly selected murder victim is male given that they are less than 18 years old is 06751

Since P(male) = 07515 and

P(male | lt 18 years old) = 06751

the events ldquomalerdquo and ldquoless than 18 years oldrdquo are not independent In fact knowing the victim is less than 18 years old decreases the probability that the victim is male

I draw one card and look

at it I tell you it is red What is the

probability it is a heart

P( heart | red) =

13P(heart and red) 152

26P(red) 252

Are ldquored cardrdquo and ldquospaderdquo mutually

exclusive Are they

independentA red card canrsquot be a spade so they ARE mutually exclusive

Are ldquored cardrdquo and

ldquoacerdquo mutually

exclusive Are they

independent2 aces are

red cards so they are

NOT mutually exclusive

Are ldquoface cardrdquo and

ldquokingrdquo mutually

exclusive Are they

independentKings are

Face cards so they are

NOT mutually exclusive

ldquoSlump I ainrsquot in no slump I just ainrsquot hittinrdquo

Yogi Berra

• PowerPoint Presentation
• Probability
• Law of Large Numbers
• Basic Rules of Probability
• Slide 5
• Slide 6
• Independent
• Slide 8
• Slide 9
• Slide 10
• Slide 11
• Slide 12
• Slide 13
• Slide 14
• Slide 15
• Slide 16
• Slide 17
• Slide 18
• Slide 19
• Slide 20
• Slide 21
• Slide 22
• Slide 23
• Slide 24
• Slide 25
• Slide 26
• Slide 27
• Slide 28
• Slide 29
• Slide 30
• Slide 31
• Slide 32
• Rule 7 Conditional Probability
• Slide 34
• Probabilities from two way tables
• Slide 36
• Slide 37
• Slide 38
• Slide 39
• Slide 40
• Slide 41
• Slide 42
• Slide 43
• Slide 44
• Slide 45
• Slide 46
• Slide 47
• Slide 48

Suppose that 40 of cars in Fort Smith are manufactured in the United States 30 in Japan 10 in Germany and 20 in other countries

If cars are selected at random what is the probability that it is not US made

P(not US made) = 1 ndash P(US made) = 1 - 4 = 6

Suppose that 40 of cars in Fort Smith are manufactured in the United States 30 in Japan 10 in Germany and 20 in other countries

If cars are selected at random what is the probability that it is made in Japan or Germany

P(Japanese or German) = P(Japanese) + P(German) = 3 + 1 = 4

Suppose that 40 of cars in Fort Smith are manufactured in the United States 30 in Japan 10 in Germany and 20 in other countries

If cars are selected at random what is the probability that you see two in a row from Japan

P(2 Japanese in a row) = P(Japanese) and P(Japanese) = P(J) x P(J) = 3 x 3 = 09

Suppose that 40 of cars in Fort Smith are manufactured in the United States 30 in Japan 10 in Germany and 20 in other countries

If cars are selected at random what is the probability that none of three cars came from Germany

P(no Germany in three) = P(not G) x P(not G) x P(not G) = 9 x 9 x 9 = 729

Suppose that 40 of cars in Fort Smith are manufactured in the United States 30 in Japan 10 in Germany and 20 in other countries

If cars are selected at random what is the probability that at least one of three cars is US made

P(at least one US in three) = 1 ndash P(no US in three) = 1 ndash (6)(6)(6) = 784

Suppose that 40 of cars in Fort Smith are manufactured in the United States 30 in Japan 10 in Germany and 20 in other countries

If cars are selected at random what is the probability that the first Japanese car is the fourth one you choose

P(first J is the fourth car) = P(not J) x P(not J) x P(not J) x P(J) = (7)3 (3) = 1029

Watch out for

bull probabilities that donrsquot add up to 1

bull donrsquot add probabilities of events if they are not disjoint

bull donrsquot multiply probabilities of events if they are not independent

bull donrsquot confuse disjoint and independent

Rule 7 Conditional Probability

bull A probability that takes into account a given condition

P(A)

B)P(AA)|P(B

P(given)

P(and)A)|P(B

In a class there are 12 boys made up of 8 Seniors and 4 Juniors There are also 8 girls made up of 3 Seniors and 5 Juniors Find the probability of choosing a boy given that he is a Senior

Boy Girl Total

Senior 8 3 11

Junior 4 5 9

Total 12 8 20

P(Boy|Senior) = 811

P(Senior|Boy) = 812 = 23

Probabilities from two way tables

Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359

12) What is the probability that the driver is a student

359195

)( StudentP

Probabilities from two way tables

Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359

13) What is the probability that the driver drives a European car

35945

)( EuropeanP

Probabilities from two way tables

Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359

14) What is the probability that the driver drives an American or Asian car

Disjoint359102212

)(

AsianorAmericanP

Probabilities from two way tables

Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359

15) What is the probability that the driver is staff or drives an Asian car

Disjoint35947102164

)(

AsianorStaffP

Probabilities from two way tables

Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359

16) What is the probability that the driver is staff and drives an Asian car

35947

)( AsianandStaffP

Probabilities from two way tables

Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359

17) If the driver is a student what is the probability that they drive an American car

Condition195107

)|( StudentAmericanP

Probabilities from two way tables

Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359

18) What is the probability that the driver is a student if the driver drives a European car

Condition 4533

)|( EuropeanStudentP

Definition of Independent Events

Two events E and F are independent if and only if

P(F | E) = P(F) or P(E | F) = P(E)

EXAMPLE Illustrating Independent Events

The probability a randomly selected murder victim is male is 07515 The probability a randomly selected murder victim is male given that they are less than 18 years old is 06751

Since P(male) = 07515 and

P(male | lt 18 years old) = 06751

the events ldquomalerdquo and ldquoless than 18 years oldrdquo are not independent In fact knowing the victim is less than 18 years old decreases the probability that the victim is male

I draw one card and look

at it I tell you it is red What is the

probability it is a heart

P( heart | red) =

13P(heart and red) 152

26P(red) 252

Are ldquored cardrdquo and ldquospaderdquo mutually

exclusive Are they

independentA red card canrsquot be a spade so they ARE mutually exclusive

Are ldquored cardrdquo and

ldquoacerdquo mutually

exclusive Are they

independent2 aces are

red cards so they are

NOT mutually exclusive

Are ldquoface cardrdquo and

ldquokingrdquo mutually

exclusive Are they

independentKings are

Face cards so they are

NOT mutually exclusive

ldquoSlump I ainrsquot in no slump I just ainrsquot hittinrdquo

Yogi Berra

• PowerPoint Presentation
• Probability
• Law of Large Numbers
• Basic Rules of Probability
• Slide 5
• Slide 6
• Independent
• Slide 8
• Slide 9
• Slide 10
• Slide 11
• Slide 12
• Slide 13
• Slide 14
• Slide 15
• Slide 16
• Slide 17
• Slide 18
• Slide 19
• Slide 20
• Slide 21
• Slide 22
• Slide 23
• Slide 24
• Slide 25
• Slide 26
• Slide 27
• Slide 28
• Slide 29
• Slide 30
• Slide 31
• Slide 32
• Rule 7 Conditional Probability
• Slide 34
• Probabilities from two way tables
• Slide 36
• Slide 37
• Slide 38
• Slide 39
• Slide 40
• Slide 41
• Slide 42
• Slide 43
• Slide 44
• Slide 45
• Slide 46
• Slide 47
• Slide 48

Suppose that 40 of cars in Fort Smith are manufactured in the United States 30 in Japan 10 in Germany and 20 in other countries

If cars are selected at random what is the probability that it is made in Japan or Germany

P(Japanese or German) = P(Japanese) + P(German) = 3 + 1 = 4

Suppose that 40 of cars in Fort Smith are manufactured in the United States 30 in Japan 10 in Germany and 20 in other countries

If cars are selected at random what is the probability that you see two in a row from Japan

P(2 Japanese in a row) = P(Japanese) and P(Japanese) = P(J) x P(J) = 3 x 3 = 09

Suppose that 40 of cars in Fort Smith are manufactured in the United States 30 in Japan 10 in Germany and 20 in other countries

If cars are selected at random what is the probability that none of three cars came from Germany

P(no Germany in three) = P(not G) x P(not G) x P(not G) = 9 x 9 x 9 = 729

Suppose that 40 of cars in Fort Smith are manufactured in the United States 30 in Japan 10 in Germany and 20 in other countries

If cars are selected at random what is the probability that at least one of three cars is US made

P(at least one US in three) = 1 ndash P(no US in three) = 1 ndash (6)(6)(6) = 784

Suppose that 40 of cars in Fort Smith are manufactured in the United States 30 in Japan 10 in Germany and 20 in other countries

If cars are selected at random what is the probability that the first Japanese car is the fourth one you choose

P(first J is the fourth car) = P(not J) x P(not J) x P(not J) x P(J) = (7)3 (3) = 1029

Watch out for

bull probabilities that donrsquot add up to 1

bull donrsquot add probabilities of events if they are not disjoint

bull donrsquot multiply probabilities of events if they are not independent

bull donrsquot confuse disjoint and independent

Rule 7 Conditional Probability

bull A probability that takes into account a given condition

P(A)

B)P(AA)|P(B

P(given)

P(and)A)|P(B

In a class there are 12 boys made up of 8 Seniors and 4 Juniors There are also 8 girls made up of 3 Seniors and 5 Juniors Find the probability of choosing a boy given that he is a Senior

Boy Girl Total

Senior 8 3 11

Junior 4 5 9

Total 12 8 20

P(Boy|Senior) = 811

P(Senior|Boy) = 812 = 23

Probabilities from two way tables

Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359

12) What is the probability that the driver is a student

359195

)( StudentP

Probabilities from two way tables

Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359

13) What is the probability that the driver drives a European car

35945

)( EuropeanP

Probabilities from two way tables

Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359

14) What is the probability that the driver drives an American or Asian car

Disjoint359102212

)(

AsianorAmericanP

Probabilities from two way tables

Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359

15) What is the probability that the driver is staff or drives an Asian car

Disjoint35947102164

)(

AsianorStaffP

Probabilities from two way tables

Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359

16) What is the probability that the driver is staff and drives an Asian car

35947

)( AsianandStaffP

Probabilities from two way tables

Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359

17) If the driver is a student what is the probability that they drive an American car

Condition195107

)|( StudentAmericanP

Probabilities from two way tables

Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359

18) What is the probability that the driver is a student if the driver drives a European car

Condition 4533

)|( EuropeanStudentP

Definition of Independent Events

Two events E and F are independent if and only if

P(F | E) = P(F) or P(E | F) = P(E)

EXAMPLE Illustrating Independent Events

The probability a randomly selected murder victim is male is 07515 The probability a randomly selected murder victim is male given that they are less than 18 years old is 06751

Since P(male) = 07515 and

P(male | lt 18 years old) = 06751

the events ldquomalerdquo and ldquoless than 18 years oldrdquo are not independent In fact knowing the victim is less than 18 years old decreases the probability that the victim is male

I draw one card and look

at it I tell you it is red What is the

probability it is a heart

P( heart | red) =

13P(heart and red) 152

26P(red) 252

Are ldquored cardrdquo and ldquospaderdquo mutually

exclusive Are they

independentA red card canrsquot be a spade so they ARE mutually exclusive

Are ldquored cardrdquo and

ldquoacerdquo mutually

exclusive Are they

independent2 aces are

red cards so they are

NOT mutually exclusive

Are ldquoface cardrdquo and

ldquokingrdquo mutually

exclusive Are they

independentKings are

Face cards so they are

NOT mutually exclusive

ldquoSlump I ainrsquot in no slump I just ainrsquot hittinrdquo

Yogi Berra

• PowerPoint Presentation
• Probability
• Law of Large Numbers
• Basic Rules of Probability
• Slide 5
• Slide 6
• Independent
• Slide 8
• Slide 9
• Slide 10
• Slide 11
• Slide 12
• Slide 13
• Slide 14
• Slide 15
• Slide 16
• Slide 17
• Slide 18
• Slide 19
• Slide 20
• Slide 21
• Slide 22
• Slide 23
• Slide 24
• Slide 25
• Slide 26
• Slide 27
• Slide 28
• Slide 29
• Slide 30
• Slide 31
• Slide 32
• Rule 7 Conditional Probability
• Slide 34
• Probabilities from two way tables
• Slide 36
• Slide 37
• Slide 38
• Slide 39
• Slide 40
• Slide 41
• Slide 42
• Slide 43
• Slide 44
• Slide 45
• Slide 46
• Slide 47
• Slide 48

Suppose that 40 of cars in Fort Smith are manufactured in the United States 30 in Japan 10 in Germany and 20 in other countries

If cars are selected at random what is the probability that you see two in a row from Japan

P(2 Japanese in a row) = P(Japanese) and P(Japanese) = P(J) x P(J) = 3 x 3 = 09

Suppose that 40 of cars in Fort Smith are manufactured in the United States 30 in Japan 10 in Germany and 20 in other countries

If cars are selected at random what is the probability that none of three cars came from Germany

P(no Germany in three) = P(not G) x P(not G) x P(not G) = 9 x 9 x 9 = 729

Suppose that 40 of cars in Fort Smith are manufactured in the United States 30 in Japan 10 in Germany and 20 in other countries

If cars are selected at random what is the probability that at least one of three cars is US made

P(at least one US in three) = 1 ndash P(no US in three) = 1 ndash (6)(6)(6) = 784

Suppose that 40 of cars in Fort Smith are manufactured in the United States 30 in Japan 10 in Germany and 20 in other countries

If cars are selected at random what is the probability that the first Japanese car is the fourth one you choose

P(first J is the fourth car) = P(not J) x P(not J) x P(not J) x P(J) = (7)3 (3) = 1029

Watch out for

bull probabilities that donrsquot add up to 1

bull donrsquot add probabilities of events if they are not disjoint

bull donrsquot multiply probabilities of events if they are not independent

bull donrsquot confuse disjoint and independent

Rule 7 Conditional Probability

bull A probability that takes into account a given condition

P(A)

B)P(AA)|P(B

P(given)

P(and)A)|P(B

In a class there are 12 boys made up of 8 Seniors and 4 Juniors There are also 8 girls made up of 3 Seniors and 5 Juniors Find the probability of choosing a boy given that he is a Senior

Boy Girl Total

Senior 8 3 11

Junior 4 5 9

Total 12 8 20

P(Boy|Senior) = 811

P(Senior|Boy) = 812 = 23

Probabilities from two way tables

Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359

12) What is the probability that the driver is a student

359195

)( StudentP

Probabilities from two way tables

Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359

13) What is the probability that the driver drives a European car

35945

)( EuropeanP

Probabilities from two way tables

Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359

14) What is the probability that the driver drives an American or Asian car

Disjoint359102212

)(

AsianorAmericanP

Probabilities from two way tables

Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359

15) What is the probability that the driver is staff or drives an Asian car

Disjoint35947102164

)(

AsianorStaffP

Probabilities from two way tables

Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359

16) What is the probability that the driver is staff and drives an Asian car

35947

)( AsianandStaffP

Probabilities from two way tables

Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359

17) If the driver is a student what is the probability that they drive an American car

Condition195107

)|( StudentAmericanP

Probabilities from two way tables

Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359

18) What is the probability that the driver is a student if the driver drives a European car

Condition 4533

)|( EuropeanStudentP

Definition of Independent Events

Two events E and F are independent if and only if

P(F | E) = P(F) or P(E | F) = P(E)

EXAMPLE Illustrating Independent Events

The probability a randomly selected murder victim is male is 07515 The probability a randomly selected murder victim is male given that they are less than 18 years old is 06751

Since P(male) = 07515 and

P(male | lt 18 years old) = 06751

the events ldquomalerdquo and ldquoless than 18 years oldrdquo are not independent In fact knowing the victim is less than 18 years old decreases the probability that the victim is male

I draw one card and look

at it I tell you it is red What is the

probability it is a heart

P( heart | red) =

13P(heart and red) 152

26P(red) 252

Are ldquored cardrdquo and ldquospaderdquo mutually

exclusive Are they

independentA red card canrsquot be a spade so they ARE mutually exclusive

Are ldquored cardrdquo and

ldquoacerdquo mutually

exclusive Are they

independent2 aces are

red cards so they are

NOT mutually exclusive

Are ldquoface cardrdquo and

ldquokingrdquo mutually

exclusive Are they

independentKings are

Face cards so they are

NOT mutually exclusive

ldquoSlump I ainrsquot in no slump I just ainrsquot hittinrdquo

Yogi Berra

• PowerPoint Presentation
• Probability
• Law of Large Numbers
• Basic Rules of Probability
• Slide 5
• Slide 6
• Independent
• Slide 8
• Slide 9
• Slide 10
• Slide 11
• Slide 12
• Slide 13
• Slide 14
• Slide 15
• Slide 16
• Slide 17
• Slide 18
• Slide 19
• Slide 20
• Slide 21
• Slide 22
• Slide 23
• Slide 24
• Slide 25
• Slide 26
• Slide 27
• Slide 28
• Slide 29
• Slide 30
• Slide 31
• Slide 32
• Rule 7 Conditional Probability
• Slide 34
• Probabilities from two way tables
• Slide 36
• Slide 37
• Slide 38
• Slide 39
• Slide 40
• Slide 41
• Slide 42
• Slide 43
• Slide 44
• Slide 45
• Slide 46
• Slide 47
• Slide 48

Suppose that 40 of cars in Fort Smith are manufactured in the United States 30 in Japan 10 in Germany and 20 in other countries

If cars are selected at random what is the probability that none of three cars came from Germany

P(no Germany in three) = P(not G) x P(not G) x P(not G) = 9 x 9 x 9 = 729

Suppose that 40 of cars in Fort Smith are manufactured in the United States 30 in Japan 10 in Germany and 20 in other countries

If cars are selected at random what is the probability that at least one of three cars is US made

P(at least one US in three) = 1 ndash P(no US in three) = 1 ndash (6)(6)(6) = 784

Suppose that 40 of cars in Fort Smith are manufactured in the United States 30 in Japan 10 in Germany and 20 in other countries

If cars are selected at random what is the probability that the first Japanese car is the fourth one you choose

P(first J is the fourth car) = P(not J) x P(not J) x P(not J) x P(J) = (7)3 (3) = 1029

Watch out for

bull probabilities that donrsquot add up to 1

bull donrsquot add probabilities of events if they are not disjoint

bull donrsquot multiply probabilities of events if they are not independent

bull donrsquot confuse disjoint and independent

Rule 7 Conditional Probability

bull A probability that takes into account a given condition

P(A)

B)P(AA)|P(B

P(given)

P(and)A)|P(B

In a class there are 12 boys made up of 8 Seniors and 4 Juniors There are also 8 girls made up of 3 Seniors and 5 Juniors Find the probability of choosing a boy given that he is a Senior

Boy Girl Total

Senior 8 3 11

Junior 4 5 9

Total 12 8 20

P(Boy|Senior) = 811

P(Senior|Boy) = 812 = 23

Probabilities from two way tables

Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359

12) What is the probability that the driver is a student

359195

)( StudentP

Probabilities from two way tables

Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359

13) What is the probability that the driver drives a European car

35945

)( EuropeanP

Probabilities from two way tables

Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359

14) What is the probability that the driver drives an American or Asian car

Disjoint359102212

)(

AsianorAmericanP

Probabilities from two way tables

Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359

15) What is the probability that the driver is staff or drives an Asian car

Disjoint35947102164

)(

AsianorStaffP

Probabilities from two way tables

Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359

16) What is the probability that the driver is staff and drives an Asian car

35947

)( AsianandStaffP

Probabilities from two way tables

Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359

17) If the driver is a student what is the probability that they drive an American car

Condition195107

)|( StudentAmericanP

Probabilities from two way tables

Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359

18) What is the probability that the driver is a student if the driver drives a European car

Condition 4533

)|( EuropeanStudentP

Definition of Independent Events

Two events E and F are independent if and only if

P(F | E) = P(F) or P(E | F) = P(E)

EXAMPLE Illustrating Independent Events

The probability a randomly selected murder victim is male is 07515 The probability a randomly selected murder victim is male given that they are less than 18 years old is 06751

Since P(male) = 07515 and

P(male | lt 18 years old) = 06751

the events ldquomalerdquo and ldquoless than 18 years oldrdquo are not independent In fact knowing the victim is less than 18 years old decreases the probability that the victim is male

I draw one card and look

at it I tell you it is red What is the

probability it is a heart

P( heart | red) =

13P(heart and red) 152

26P(red) 252

Are ldquored cardrdquo and ldquospaderdquo mutually

exclusive Are they

independentA red card canrsquot be a spade so they ARE mutually exclusive

Are ldquored cardrdquo and

ldquoacerdquo mutually

exclusive Are they

independent2 aces are

red cards so they are

NOT mutually exclusive

Are ldquoface cardrdquo and

ldquokingrdquo mutually

exclusive Are they

independentKings are

Face cards so they are

NOT mutually exclusive

ldquoSlump I ainrsquot in no slump I just ainrsquot hittinrdquo

Yogi Berra

• PowerPoint Presentation
• Probability
• Law of Large Numbers
• Basic Rules of Probability
• Slide 5
• Slide 6
• Independent
• Slide 8
• Slide 9
• Slide 10
• Slide 11
• Slide 12
• Slide 13
• Slide 14
• Slide 15
• Slide 16
• Slide 17
• Slide 18
• Slide 19
• Slide 20
• Slide 21
• Slide 22
• Slide 23
• Slide 24
• Slide 25
• Slide 26
• Slide 27
• Slide 28
• Slide 29
• Slide 30
• Slide 31
• Slide 32
• Rule 7 Conditional Probability
• Slide 34
• Probabilities from two way tables
• Slide 36
• Slide 37
• Slide 38
• Slide 39
• Slide 40
• Slide 41
• Slide 42
• Slide 43
• Slide 44
• Slide 45
• Slide 46
• Slide 47
• Slide 48

Suppose that 40 of cars in Fort Smith are manufactured in the United States 30 in Japan 10 in Germany and 20 in other countries

If cars are selected at random what is the probability that at least one of three cars is US made

P(at least one US in three) = 1 ndash P(no US in three) = 1 ndash (6)(6)(6) = 784

Suppose that 40 of cars in Fort Smith are manufactured in the United States 30 in Japan 10 in Germany and 20 in other countries

If cars are selected at random what is the probability that the first Japanese car is the fourth one you choose

P(first J is the fourth car) = P(not J) x P(not J) x P(not J) x P(J) = (7)3 (3) = 1029

Watch out for

bull probabilities that donrsquot add up to 1

bull donrsquot add probabilities of events if they are not disjoint

bull donrsquot multiply probabilities of events if they are not independent

bull donrsquot confuse disjoint and independent

Rule 7 Conditional Probability

bull A probability that takes into account a given condition

P(A)

B)P(AA)|P(B

P(given)

P(and)A)|P(B

In a class there are 12 boys made up of 8 Seniors and 4 Juniors There are also 8 girls made up of 3 Seniors and 5 Juniors Find the probability of choosing a boy given that he is a Senior

Boy Girl Total

Senior 8 3 11

Junior 4 5 9

Total 12 8 20

P(Boy|Senior) = 811

P(Senior|Boy) = 812 = 23

Probabilities from two way tables

Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359

12) What is the probability that the driver is a student

359195

)( StudentP

Probabilities from two way tables

Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359

13) What is the probability that the driver drives a European car

35945

)( EuropeanP

Probabilities from two way tables

Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359

14) What is the probability that the driver drives an American or Asian car

Disjoint359102212

)(

AsianorAmericanP

Probabilities from two way tables

Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359

15) What is the probability that the driver is staff or drives an Asian car

Disjoint35947102164

)(

AsianorStaffP

Probabilities from two way tables

Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359

16) What is the probability that the driver is staff and drives an Asian car

35947

)( AsianandStaffP

Probabilities from two way tables

Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359

17) If the driver is a student what is the probability that they drive an American car

Condition195107

)|( StudentAmericanP

Probabilities from two way tables

Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359

18) What is the probability that the driver is a student if the driver drives a European car

Condition 4533

)|( EuropeanStudentP

Definition of Independent Events

Two events E and F are independent if and only if

P(F | E) = P(F) or P(E | F) = P(E)

EXAMPLE Illustrating Independent Events

The probability a randomly selected murder victim is male is 07515 The probability a randomly selected murder victim is male given that they are less than 18 years old is 06751

Since P(male) = 07515 and

P(male | lt 18 years old) = 06751

the events ldquomalerdquo and ldquoless than 18 years oldrdquo are not independent In fact knowing the victim is less than 18 years old decreases the probability that the victim is male

I draw one card and look

at it I tell you it is red What is the

probability it is a heart

P( heart | red) =

13P(heart and red) 152

26P(red) 252

Are ldquored cardrdquo and ldquospaderdquo mutually

exclusive Are they

independentA red card canrsquot be a spade so they ARE mutually exclusive

Are ldquored cardrdquo and

ldquoacerdquo mutually

exclusive Are they

independent2 aces are

red cards so they are

NOT mutually exclusive

Are ldquoface cardrdquo and

ldquokingrdquo mutually

exclusive Are they

independentKings are

Face cards so they are

NOT mutually exclusive

ldquoSlump I ainrsquot in no slump I just ainrsquot hittinrdquo

Yogi Berra

• PowerPoint Presentation
• Probability
• Law of Large Numbers
• Basic Rules of Probability
• Slide 5
• Slide 6
• Independent
• Slide 8
• Slide 9
• Slide 10
• Slide 11
• Slide 12
• Slide 13
• Slide 14
• Slide 15
• Slide 16
• Slide 17
• Slide 18
• Slide 19
• Slide 20
• Slide 21
• Slide 22
• Slide 23
• Slide 24
• Slide 25
• Slide 26
• Slide 27
• Slide 28
• Slide 29
• Slide 30
• Slide 31
• Slide 32
• Rule 7 Conditional Probability
• Slide 34
• Probabilities from two way tables
• Slide 36
• Slide 37
• Slide 38
• Slide 39
• Slide 40
• Slide 41
• Slide 42
• Slide 43
• Slide 44
• Slide 45
• Slide 46
• Slide 47
• Slide 48

Suppose that 40 of cars in Fort Smith are manufactured in the United States 30 in Japan 10 in Germany and 20 in other countries

If cars are selected at random what is the probability that the first Japanese car is the fourth one you choose

P(first J is the fourth car) = P(not J) x P(not J) x P(not J) x P(J) = (7)3 (3) = 1029

Watch out for

bull probabilities that donrsquot add up to 1

bull donrsquot add probabilities of events if they are not disjoint

bull donrsquot multiply probabilities of events if they are not independent

bull donrsquot confuse disjoint and independent

Rule 7 Conditional Probability

bull A probability that takes into account a given condition

P(A)

B)P(AA)|P(B

P(given)

P(and)A)|P(B

In a class there are 12 boys made up of 8 Seniors and 4 Juniors There are also 8 girls made up of 3 Seniors and 5 Juniors Find the probability of choosing a boy given that he is a Senior

Boy Girl Total

Senior 8 3 11

Junior 4 5 9

Total 12 8 20

P(Boy|Senior) = 811

P(Senior|Boy) = 812 = 23

Probabilities from two way tables

Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359

12) What is the probability that the driver is a student

359195

)( StudentP

Probabilities from two way tables

Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359

13) What is the probability that the driver drives a European car

35945

)( EuropeanP

Probabilities from two way tables

Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359

14) What is the probability that the driver drives an American or Asian car

Disjoint359102212

)(

AsianorAmericanP

Probabilities from two way tables

Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359

15) What is the probability that the driver is staff or drives an Asian car

Disjoint35947102164

)(

AsianorStaffP

Probabilities from two way tables

Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359

16) What is the probability that the driver is staff and drives an Asian car

35947

)( AsianandStaffP

Probabilities from two way tables

Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359

17) If the driver is a student what is the probability that they drive an American car

Condition195107

)|( StudentAmericanP

Probabilities from two way tables

Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359

18) What is the probability that the driver is a student if the driver drives a European car

Condition 4533

)|( EuropeanStudentP

Definition of Independent Events

Two events E and F are independent if and only if

P(F | E) = P(F) or P(E | F) = P(E)

EXAMPLE Illustrating Independent Events

The probability a randomly selected murder victim is male is 07515 The probability a randomly selected murder victim is male given that they are less than 18 years old is 06751

Since P(male) = 07515 and

P(male | lt 18 years old) = 06751

the events ldquomalerdquo and ldquoless than 18 years oldrdquo are not independent In fact knowing the victim is less than 18 years old decreases the probability that the victim is male

I draw one card and look

at it I tell you it is red What is the

probability it is a heart

P( heart | red) =

13P(heart and red) 152

26P(red) 252

Are ldquored cardrdquo and ldquospaderdquo mutually

exclusive Are they

independentA red card canrsquot be a spade so they ARE mutually exclusive

Are ldquored cardrdquo and

ldquoacerdquo mutually

exclusive Are they

independent2 aces are

red cards so they are

NOT mutually exclusive

Are ldquoface cardrdquo and

ldquokingrdquo mutually

exclusive Are they

independentKings are

Face cards so they are

NOT mutually exclusive

ldquoSlump I ainrsquot in no slump I just ainrsquot hittinrdquo

Yogi Berra

• PowerPoint Presentation
• Probability
• Law of Large Numbers
• Basic Rules of Probability
• Slide 5
• Slide 6
• Independent
• Slide 8
• Slide 9
• Slide 10
• Slide 11
• Slide 12
• Slide 13
• Slide 14
• Slide 15
• Slide 16
• Slide 17
• Slide 18
• Slide 19
• Slide 20
• Slide 21
• Slide 22
• Slide 23
• Slide 24
• Slide 25
• Slide 26
• Slide 27
• Slide 28
• Slide 29
• Slide 30
• Slide 31
• Slide 32
• Rule 7 Conditional Probability
• Slide 34
• Probabilities from two way tables
• Slide 36
• Slide 37
• Slide 38
• Slide 39
• Slide 40
• Slide 41
• Slide 42
• Slide 43
• Slide 44
• Slide 45
• Slide 46
• Slide 47
• Slide 48

Watch out for

bull probabilities that donrsquot add up to 1

bull donrsquot add probabilities of events if they are not disjoint

bull donrsquot multiply probabilities of events if they are not independent

bull donrsquot confuse disjoint and independent

Rule 7 Conditional Probability

bull A probability that takes into account a given condition

P(A)

B)P(AA)|P(B

P(given)

P(and)A)|P(B

In a class there are 12 boys made up of 8 Seniors and 4 Juniors There are also 8 girls made up of 3 Seniors and 5 Juniors Find the probability of choosing a boy given that he is a Senior

Boy Girl Total

Senior 8 3 11

Junior 4 5 9

Total 12 8 20

P(Boy|Senior) = 811

P(Senior|Boy) = 812 = 23

Probabilities from two way tables

Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359

12) What is the probability that the driver is a student

359195

)( StudentP

Probabilities from two way tables

Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359

13) What is the probability that the driver drives a European car

35945

)( EuropeanP

Probabilities from two way tables

Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359

14) What is the probability that the driver drives an American or Asian car

Disjoint359102212

)(

AsianorAmericanP

Probabilities from two way tables

Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359

15) What is the probability that the driver is staff or drives an Asian car

Disjoint35947102164

)(

AsianorStaffP

Probabilities from two way tables

Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359

16) What is the probability that the driver is staff and drives an Asian car

35947

)( AsianandStaffP

Probabilities from two way tables

Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359

17) If the driver is a student what is the probability that they drive an American car

Condition195107

)|( StudentAmericanP

Probabilities from two way tables

Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359

18) What is the probability that the driver is a student if the driver drives a European car

Condition 4533

)|( EuropeanStudentP

Definition of Independent Events

Two events E and F are independent if and only if

P(F | E) = P(F) or P(E | F) = P(E)

EXAMPLE Illustrating Independent Events

The probability a randomly selected murder victim is male is 07515 The probability a randomly selected murder victim is male given that they are less than 18 years old is 06751

Since P(male) = 07515 and

P(male | lt 18 years old) = 06751

the events ldquomalerdquo and ldquoless than 18 years oldrdquo are not independent In fact knowing the victim is less than 18 years old decreases the probability that the victim is male

I draw one card and look

at it I tell you it is red What is the

probability it is a heart

P( heart | red) =

13P(heart and red) 152

26P(red) 252

Are ldquored cardrdquo and ldquospaderdquo mutually

exclusive Are they

independentA red card canrsquot be a spade so they ARE mutually exclusive

Are ldquored cardrdquo and

ldquoacerdquo mutually

exclusive Are they

independent2 aces are

red cards so they are

NOT mutually exclusive

Are ldquoface cardrdquo and

ldquokingrdquo mutually

exclusive Are they

independentKings are

Face cards so they are

NOT mutually exclusive

ldquoSlump I ainrsquot in no slump I just ainrsquot hittinrdquo

Yogi Berra

• PowerPoint Presentation
• Probability
• Law of Large Numbers
• Basic Rules of Probability
• Slide 5
• Slide 6
• Independent
• Slide 8
• Slide 9
• Slide 10
• Slide 11
• Slide 12
• Slide 13
• Slide 14
• Slide 15
• Slide 16
• Slide 17
• Slide 18
• Slide 19
• Slide 20
• Slide 21
• Slide 22
• Slide 23
• Slide 24
• Slide 25
• Slide 26
• Slide 27
• Slide 28
• Slide 29
• Slide 30
• Slide 31
• Slide 32
• Rule 7 Conditional Probability
• Slide 34
• Probabilities from two way tables
• Slide 36
• Slide 37
• Slide 38
• Slide 39
• Slide 40
• Slide 41
• Slide 42
• Slide 43
• Slide 44
• Slide 45
• Slide 46
• Slide 47
• Slide 48

Rule 7 Conditional Probability

bull A probability that takes into account a given condition

P(A)

B)P(AA)|P(B

P(given)

P(and)A)|P(B

In a class there are 12 boys made up of 8 Seniors and 4 Juniors There are also 8 girls made up of 3 Seniors and 5 Juniors Find the probability of choosing a boy given that he is a Senior

Boy Girl Total

Senior 8 3 11

Junior 4 5 9

Total 12 8 20

P(Boy|Senior) = 811

P(Senior|Boy) = 812 = 23

Probabilities from two way tables

Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359

12) What is the probability that the driver is a student

359195

)( StudentP

Probabilities from two way tables

Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359

13) What is the probability that the driver drives a European car

35945

)( EuropeanP

Probabilities from two way tables

Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359

14) What is the probability that the driver drives an American or Asian car

Disjoint359102212

)(

AsianorAmericanP

Probabilities from two way tables

Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359

15) What is the probability that the driver is staff or drives an Asian car

Disjoint35947102164

)(

AsianorStaffP

Probabilities from two way tables

Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359

16) What is the probability that the driver is staff and drives an Asian car

35947

)( AsianandStaffP

Probabilities from two way tables

Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359

17) If the driver is a student what is the probability that they drive an American car

Condition195107

)|( StudentAmericanP

Probabilities from two way tables

Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359

18) What is the probability that the driver is a student if the driver drives a European car

Condition 4533

)|( EuropeanStudentP

Definition of Independent Events

Two events E and F are independent if and only if

P(F | E) = P(F) or P(E | F) = P(E)

EXAMPLE Illustrating Independent Events

The probability a randomly selected murder victim is male is 07515 The probability a randomly selected murder victim is male given that they are less than 18 years old is 06751

Since P(male) = 07515 and

P(male | lt 18 years old) = 06751

the events ldquomalerdquo and ldquoless than 18 years oldrdquo are not independent In fact knowing the victim is less than 18 years old decreases the probability that the victim is male

I draw one card and look

at it I tell you it is red What is the

probability it is a heart

P( heart | red) =

13P(heart and red) 152

26P(red) 252

Are ldquored cardrdquo and ldquospaderdquo mutually

exclusive Are they

independentA red card canrsquot be a spade so they ARE mutually exclusive

Are ldquored cardrdquo and

ldquoacerdquo mutually

exclusive Are they

independent2 aces are

red cards so they are

NOT mutually exclusive

Are ldquoface cardrdquo and

ldquokingrdquo mutually

exclusive Are they

independentKings are

Face cards so they are

NOT mutually exclusive

ldquoSlump I ainrsquot in no slump I just ainrsquot hittinrdquo

Yogi Berra

• PowerPoint Presentation
• Probability
• Law of Large Numbers
• Basic Rules of Probability
• Slide 5
• Slide 6
• Independent
• Slide 8
• Slide 9
• Slide 10
• Slide 11
• Slide 12
• Slide 13
• Slide 14
• Slide 15
• Slide 16
• Slide 17
• Slide 18
• Slide 19
• Slide 20
• Slide 21
• Slide 22
• Slide 23
• Slide 24
• Slide 25
• Slide 26
• Slide 27
• Slide 28
• Slide 29
• Slide 30
• Slide 31
• Slide 32
• Rule 7 Conditional Probability
• Slide 34
• Probabilities from two way tables
• Slide 36
• Slide 37
• Slide 38
• Slide 39
• Slide 40
• Slide 41
• Slide 42
• Slide 43
• Slide 44
• Slide 45
• Slide 46
• Slide 47
• Slide 48

In a class there are 12 boys made up of 8 Seniors and 4 Juniors There are also 8 girls made up of 3 Seniors and 5 Juniors Find the probability of choosing a boy given that he is a Senior

Boy Girl Total

Senior 8 3 11

Junior 4 5 9

Total 12 8 20

P(Boy|Senior) = 811

P(Senior|Boy) = 812 = 23

Probabilities from two way tables

Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359

12) What is the probability that the driver is a student

359195

)( StudentP

Probabilities from two way tables

Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359

13) What is the probability that the driver drives a European car

35945

)( EuropeanP

Probabilities from two way tables

Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359

14) What is the probability that the driver drives an American or Asian car

Disjoint359102212

)(

AsianorAmericanP

Probabilities from two way tables

Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359

15) What is the probability that the driver is staff or drives an Asian car

Disjoint35947102164

)(

AsianorStaffP

Probabilities from two way tables

Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359

16) What is the probability that the driver is staff and drives an Asian car

35947

)( AsianandStaffP

Probabilities from two way tables

Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359

17) If the driver is a student what is the probability that they drive an American car

Condition195107

)|( StudentAmericanP

Probabilities from two way tables

Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359

18) What is the probability that the driver is a student if the driver drives a European car

Condition 4533

)|( EuropeanStudentP

Definition of Independent Events

Two events E and F are independent if and only if

P(F | E) = P(F) or P(E | F) = P(E)

EXAMPLE Illustrating Independent Events

The probability a randomly selected murder victim is male is 07515 The probability a randomly selected murder victim is male given that they are less than 18 years old is 06751

Since P(male) = 07515 and

P(male | lt 18 years old) = 06751

the events ldquomalerdquo and ldquoless than 18 years oldrdquo are not independent In fact knowing the victim is less than 18 years old decreases the probability that the victim is male

I draw one card and look

at it I tell you it is red What is the

probability it is a heart

P( heart | red) =

13P(heart and red) 152

26P(red) 252

Are ldquored cardrdquo and ldquospaderdquo mutually

exclusive Are they

independentA red card canrsquot be a spade so they ARE mutually exclusive

Are ldquored cardrdquo and

ldquoacerdquo mutually

exclusive Are they

independent2 aces are

red cards so they are

NOT mutually exclusive

Are ldquoface cardrdquo and

ldquokingrdquo mutually

exclusive Are they

independentKings are

Face cards so they are

NOT mutually exclusive

ldquoSlump I ainrsquot in no slump I just ainrsquot hittinrdquo

Yogi Berra

• PowerPoint Presentation
• Probability
• Law of Large Numbers
• Basic Rules of Probability
• Slide 5
• Slide 6
• Independent
• Slide 8
• Slide 9
• Slide 10
• Slide 11
• Slide 12
• Slide 13
• Slide 14
• Slide 15
• Slide 16
• Slide 17
• Slide 18
• Slide 19
• Slide 20
• Slide 21
• Slide 22
• Slide 23
• Slide 24
• Slide 25
• Slide 26
• Slide 27
• Slide 28
• Slide 29
• Slide 30
• Slide 31
• Slide 32
• Rule 7 Conditional Probability
• Slide 34
• Probabilities from two way tables
• Slide 36
• Slide 37
• Slide 38
• Slide 39
• Slide 40
• Slide 41
• Slide 42
• Slide 43
• Slide 44
• Slide 45
• Slide 46
• Slide 47
• Slide 48

Probabilities from two way tables

Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359

12) What is the probability that the driver is a student

359195

)( StudentP

Probabilities from two way tables

Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359

13) What is the probability that the driver drives a European car

35945

)( EuropeanP

Probabilities from two way tables

Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359

14) What is the probability that the driver drives an American or Asian car

Disjoint359102212

)(

AsianorAmericanP

Probabilities from two way tables

Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359

15) What is the probability that the driver is staff or drives an Asian car

Disjoint35947102164

)(

AsianorStaffP

Probabilities from two way tables

Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359

16) What is the probability that the driver is staff and drives an Asian car

35947

)( AsianandStaffP

Probabilities from two way tables

Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359

17) If the driver is a student what is the probability that they drive an American car

Condition195107

)|( StudentAmericanP

Probabilities from two way tables

Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359

18) What is the probability that the driver is a student if the driver drives a European car

Condition 4533

)|( EuropeanStudentP

Definition of Independent Events

Two events E and F are independent if and only if

P(F | E) = P(F) or P(E | F) = P(E)

EXAMPLE Illustrating Independent Events

The probability a randomly selected murder victim is male is 07515 The probability a randomly selected murder victim is male given that they are less than 18 years old is 06751

Since P(male) = 07515 and

P(male | lt 18 years old) = 06751

the events ldquomalerdquo and ldquoless than 18 years oldrdquo are not independent In fact knowing the victim is less than 18 years old decreases the probability that the victim is male

I draw one card and look

at it I tell you it is red What is the

probability it is a heart

P( heart | red) =

13P(heart and red) 152

26P(red) 252

Are ldquored cardrdquo and ldquospaderdquo mutually

exclusive Are they

independentA red card canrsquot be a spade so they ARE mutually exclusive

Are ldquored cardrdquo and

ldquoacerdquo mutually

exclusive Are they

independent2 aces are

red cards so they are

NOT mutually exclusive

Are ldquoface cardrdquo and

ldquokingrdquo mutually

exclusive Are they

independentKings are

Face cards so they are

NOT mutually exclusive

ldquoSlump I ainrsquot in no slump I just ainrsquot hittinrdquo

Yogi Berra

• PowerPoint Presentation
• Probability
• Law of Large Numbers
• Basic Rules of Probability
• Slide 5
• Slide 6
• Independent
• Slide 8
• Slide 9
• Slide 10
• Slide 11
• Slide 12
• Slide 13
• Slide 14
• Slide 15
• Slide 16
• Slide 17
• Slide 18
• Slide 19
• Slide 20
• Slide 21
• Slide 22
• Slide 23
• Slide 24
• Slide 25
• Slide 26
• Slide 27
• Slide 28
• Slide 29
• Slide 30
• Slide 31
• Slide 32
• Rule 7 Conditional Probability
• Slide 34
• Probabilities from two way tables
• Slide 36
• Slide 37
• Slide 38
• Slide 39
• Slide 40
• Slide 41
• Slide 42
• Slide 43
• Slide 44
• Slide 45
• Slide 46
• Slide 47
• Slide 48

Probabilities from two way tables

Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359

13) What is the probability that the driver drives a European car

35945

)( EuropeanP

Probabilities from two way tables

Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359

14) What is the probability that the driver drives an American or Asian car

Disjoint359102212

)(

AsianorAmericanP

Probabilities from two way tables

Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359

15) What is the probability that the driver is staff or drives an Asian car

Disjoint35947102164

)(

AsianorStaffP

Probabilities from two way tables

Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359

16) What is the probability that the driver is staff and drives an Asian car

35947

)( AsianandStaffP

Probabilities from two way tables

Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359

17) If the driver is a student what is the probability that they drive an American car

Condition195107

)|( StudentAmericanP

Probabilities from two way tables

Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359

18) What is the probability that the driver is a student if the driver drives a European car

Condition 4533

)|( EuropeanStudentP

Definition of Independent Events

Two events E and F are independent if and only if

P(F | E) = P(F) or P(E | F) = P(E)

EXAMPLE Illustrating Independent Events

The probability a randomly selected murder victim is male is 07515 The probability a randomly selected murder victim is male given that they are less than 18 years old is 06751

Since P(male) = 07515 and

P(male | lt 18 years old) = 06751

the events ldquomalerdquo and ldquoless than 18 years oldrdquo are not independent In fact knowing the victim is less than 18 years old decreases the probability that the victim is male

I draw one card and look

at it I tell you it is red What is the

probability it is a heart

P( heart | red) =

13P(heart and red) 152

26P(red) 252

Are ldquored cardrdquo and ldquospaderdquo mutually

exclusive Are they

independentA red card canrsquot be a spade so they ARE mutually exclusive

Are ldquored cardrdquo and

ldquoacerdquo mutually

exclusive Are they

independent2 aces are

red cards so they are

NOT mutually exclusive

Are ldquoface cardrdquo and

ldquokingrdquo mutually

exclusive Are they

independentKings are

Face cards so they are

NOT mutually exclusive

ldquoSlump I ainrsquot in no slump I just ainrsquot hittinrdquo

Yogi Berra

• PowerPoint Presentation
• Probability
• Law of Large Numbers
• Basic Rules of Probability
• Slide 5
• Slide 6
• Independent
• Slide 8
• Slide 9
• Slide 10
• Slide 11
• Slide 12
• Slide 13
• Slide 14
• Slide 15
• Slide 16
• Slide 17
• Slide 18
• Slide 19
• Slide 20
• Slide 21
• Slide 22
• Slide 23
• Slide 24
• Slide 25
• Slide 26
• Slide 27
• Slide 28
• Slide 29
• Slide 30
• Slide 31
• Slide 32
• Rule 7 Conditional Probability
• Slide 34
• Probabilities from two way tables
• Slide 36
• Slide 37
• Slide 38
• Slide 39
• Slide 40
• Slide 41
• Slide 42
• Slide 43
• Slide 44
• Slide 45
• Slide 46
• Slide 47
• Slide 48

Probabilities from two way tables

Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359

14) What is the probability that the driver drives an American or Asian car

Disjoint359102212

)(

AsianorAmericanP

Probabilities from two way tables

Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359

15) What is the probability that the driver is staff or drives an Asian car

Disjoint35947102164

)(

AsianorStaffP

Probabilities from two way tables

Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359

16) What is the probability that the driver is staff and drives an Asian car

35947

)( AsianandStaffP

Probabilities from two way tables

Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359

17) If the driver is a student what is the probability that they drive an American car

Condition195107

)|( StudentAmericanP

Probabilities from two way tables

Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359

18) What is the probability that the driver is a student if the driver drives a European car

Condition 4533

)|( EuropeanStudentP

Definition of Independent Events

Two events E and F are independent if and only if

P(F | E) = P(F) or P(E | F) = P(E)

EXAMPLE Illustrating Independent Events

The probability a randomly selected murder victim is male is 07515 The probability a randomly selected murder victim is male given that they are less than 18 years old is 06751

Since P(male) = 07515 and

P(male | lt 18 years old) = 06751

the events ldquomalerdquo and ldquoless than 18 years oldrdquo are not independent In fact knowing the victim is less than 18 years old decreases the probability that the victim is male

I draw one card and look

at it I tell you it is red What is the

probability it is a heart

P( heart | red) =

13P(heart and red) 152

26P(red) 252

Are ldquored cardrdquo and ldquospaderdquo mutually

exclusive Are they

independentA red card canrsquot be a spade so they ARE mutually exclusive

Are ldquored cardrdquo and

ldquoacerdquo mutually

exclusive Are they

independent2 aces are

red cards so they are

NOT mutually exclusive

Are ldquoface cardrdquo and

ldquokingrdquo mutually

exclusive Are they

independentKings are

Face cards so they are

NOT mutually exclusive

ldquoSlump I ainrsquot in no slump I just ainrsquot hittinrdquo

Yogi Berra

• PowerPoint Presentation
• Probability
• Law of Large Numbers
• Basic Rules of Probability
• Slide 5
• Slide 6
• Independent
• Slide 8
• Slide 9
• Slide 10
• Slide 11
• Slide 12
• Slide 13
• Slide 14
• Slide 15
• Slide 16
• Slide 17
• Slide 18
• Slide 19
• Slide 20
• Slide 21
• Slide 22
• Slide 23
• Slide 24
• Slide 25
• Slide 26
• Slide 27
• Slide 28
• Slide 29
• Slide 30
• Slide 31
• Slide 32
• Rule 7 Conditional Probability
• Slide 34
• Probabilities from two way tables
• Slide 36
• Slide 37
• Slide 38
• Slide 39
• Slide 40
• Slide 41
• Slide 42
• Slide 43
• Slide 44
• Slide 45
• Slide 46
• Slide 47
• Slide 48

Probabilities from two way tables

Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359

15) What is the probability that the driver is staff or drives an Asian car

Disjoint35947102164

)(

AsianorStaffP

Probabilities from two way tables

Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359

16) What is the probability that the driver is staff and drives an Asian car

35947

)( AsianandStaffP

Probabilities from two way tables

Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359

17) If the driver is a student what is the probability that they drive an American car

Condition195107

)|( StudentAmericanP

Probabilities from two way tables

Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359

18) What is the probability that the driver is a student if the driver drives a European car

Condition 4533

)|( EuropeanStudentP

Definition of Independent Events

Two events E and F are independent if and only if

P(F | E) = P(F) or P(E | F) = P(E)

EXAMPLE Illustrating Independent Events

The probability a randomly selected murder victim is male is 07515 The probability a randomly selected murder victim is male given that they are less than 18 years old is 06751

Since P(male) = 07515 and

P(male | lt 18 years old) = 06751

the events ldquomalerdquo and ldquoless than 18 years oldrdquo are not independent In fact knowing the victim is less than 18 years old decreases the probability that the victim is male

I draw one card and look

at it I tell you it is red What is the

probability it is a heart

P( heart | red) =

13P(heart and red) 152

26P(red) 252

Are ldquored cardrdquo and ldquospaderdquo mutually

exclusive Are they

independentA red card canrsquot be a spade so they ARE mutually exclusive

Are ldquored cardrdquo and

ldquoacerdquo mutually

exclusive Are they

independent2 aces are

red cards so they are

NOT mutually exclusive

Are ldquoface cardrdquo and

ldquokingrdquo mutually

exclusive Are they

independentKings are

Face cards so they are

NOT mutually exclusive

ldquoSlump I ainrsquot in no slump I just ainrsquot hittinrdquo

Yogi Berra

• PowerPoint Presentation
• Probability
• Law of Large Numbers
• Basic Rules of Probability
• Slide 5
• Slide 6
• Independent
• Slide 8
• Slide 9
• Slide 10
• Slide 11
• Slide 12
• Slide 13
• Slide 14
• Slide 15
• Slide 16
• Slide 17
• Slide 18
• Slide 19
• Slide 20
• Slide 21
• Slide 22
• Slide 23
• Slide 24
• Slide 25
• Slide 26
• Slide 27
• Slide 28
• Slide 29
• Slide 30
• Slide 31
• Slide 32
• Rule 7 Conditional Probability
• Slide 34
• Probabilities from two way tables
• Slide 36
• Slide 37
• Slide 38
• Slide 39
• Slide 40
• Slide 41
• Slide 42
• Slide 43
• Slide 44
• Slide 45
• Slide 46
• Slide 47
• Slide 48

Probabilities from two way tables

Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359

16) What is the probability that the driver is staff and drives an Asian car

35947

)( AsianandStaffP

Probabilities from two way tables

Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359

17) If the driver is a student what is the probability that they drive an American car

Condition195107

)|( StudentAmericanP

Probabilities from two way tables

Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359

18) What is the probability that the driver is a student if the driver drives a European car

Condition 4533

)|( EuropeanStudentP

Definition of Independent Events

Two events E and F are independent if and only if

P(F | E) = P(F) or P(E | F) = P(E)

EXAMPLE Illustrating Independent Events

The probability a randomly selected murder victim is male is 07515 The probability a randomly selected murder victim is male given that they are less than 18 years old is 06751

Since P(male) = 07515 and

P(male | lt 18 years old) = 06751

the events ldquomalerdquo and ldquoless than 18 years oldrdquo are not independent In fact knowing the victim is less than 18 years old decreases the probability that the victim is male

I draw one card and look

at it I tell you it is red What is the

probability it is a heart

P( heart | red) =

13P(heart and red) 152

26P(red) 252

Are ldquored cardrdquo and ldquospaderdquo mutually

exclusive Are they

independentA red card canrsquot be a spade so they ARE mutually exclusive

Are ldquored cardrdquo and

ldquoacerdquo mutually

exclusive Are they

independent2 aces are

red cards so they are

NOT mutually exclusive

Are ldquoface cardrdquo and

ldquokingrdquo mutually

exclusive Are they

independentKings are

Face cards so they are

NOT mutually exclusive

ldquoSlump I ainrsquot in no slump I just ainrsquot hittinrdquo

Yogi Berra

• PowerPoint Presentation
• Probability
• Law of Large Numbers
• Basic Rules of Probability
• Slide 5
• Slide 6
• Independent
• Slide 8
• Slide 9
• Slide 10
• Slide 11
• Slide 12
• Slide 13
• Slide 14
• Slide 15
• Slide 16
• Slide 17
• Slide 18
• Slide 19
• Slide 20
• Slide 21
• Slide 22
• Slide 23
• Slide 24
• Slide 25
• Slide 26
• Slide 27
• Slide 28
• Slide 29
• Slide 30
• Slide 31
• Slide 32
• Rule 7 Conditional Probability
• Slide 34
• Probabilities from two way tables
• Slide 36
• Slide 37
• Slide 38
• Slide 39
• Slide 40
• Slide 41
• Slide 42
• Slide 43
• Slide 44
• Slide 45
• Slide 46
• Slide 47
• Slide 48

Probabilities from two way tables

Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359

17) If the driver is a student what is the probability that they drive an American car

Condition195107

)|( StudentAmericanP

Probabilities from two way tables

Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359

18) What is the probability that the driver is a student if the driver drives a European car

Condition 4533

)|( EuropeanStudentP

Definition of Independent Events

Two events E and F are independent if and only if

P(F | E) = P(F) or P(E | F) = P(E)

EXAMPLE Illustrating Independent Events

The probability a randomly selected murder victim is male is 07515 The probability a randomly selected murder victim is male given that they are less than 18 years old is 06751

Since P(male) = 07515 and

P(male | lt 18 years old) = 06751

the events ldquomalerdquo and ldquoless than 18 years oldrdquo are not independent In fact knowing the victim is less than 18 years old decreases the probability that the victim is male

I draw one card and look

at it I tell you it is red What is the

probability it is a heart

P( heart | red) =

13P(heart and red) 152

26P(red) 252

Are ldquored cardrdquo and ldquospaderdquo mutually

exclusive Are they

independentA red card canrsquot be a spade so they ARE mutually exclusive

Are ldquored cardrdquo and

ldquoacerdquo mutually

exclusive Are they

independent2 aces are

red cards so they are

NOT mutually exclusive

Are ldquoface cardrdquo and

ldquokingrdquo mutually

exclusive Are they

independentKings are

Face cards so they are

NOT mutually exclusive

ldquoSlump I ainrsquot in no slump I just ainrsquot hittinrdquo

Yogi Berra

• PowerPoint Presentation
• Probability
• Law of Large Numbers
• Basic Rules of Probability
• Slide 5
• Slide 6
• Independent
• Slide 8
• Slide 9
• Slide 10
• Slide 11
• Slide 12
• Slide 13
• Slide 14
• Slide 15
• Slide 16
• Slide 17
• Slide 18
• Slide 19
• Slide 20
• Slide 21
• Slide 22
• Slide 23
• Slide 24
• Slide 25
• Slide 26
• Slide 27
• Slide 28
• Slide 29
• Slide 30
• Slide 31
• Slide 32
• Rule 7 Conditional Probability
• Slide 34
• Probabilities from two way tables
• Slide 36
• Slide 37
• Slide 38
• Slide 39
• Slide 40
• Slide 41
• Slide 42
• Slide 43
• Slide 44
• Slide 45
• Slide 46
• Slide 47
• Slide 48

Probabilities from two way tables

Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359

18) What is the probability that the driver is a student if the driver drives a European car

Condition 4533

)|( EuropeanStudentP

Definition of Independent Events

Two events E and F are independent if and only if

P(F | E) = P(F) or P(E | F) = P(E)

EXAMPLE Illustrating Independent Events

The probability a randomly selected murder victim is male is 07515 The probability a randomly selected murder victim is male given that they are less than 18 years old is 06751

Since P(male) = 07515 and

P(male | lt 18 years old) = 06751

the events ldquomalerdquo and ldquoless than 18 years oldrdquo are not independent In fact knowing the victim is less than 18 years old decreases the probability that the victim is male

I draw one card and look

at it I tell you it is red What is the

probability it is a heart

P( heart | red) =

13P(heart and red) 152

26P(red) 252

Are ldquored cardrdquo and ldquospaderdquo mutually

exclusive Are they

independentA red card canrsquot be a spade so they ARE mutually exclusive

Are ldquored cardrdquo and

ldquoacerdquo mutually

exclusive Are they

independent2 aces are

red cards so they are

NOT mutually exclusive

Are ldquoface cardrdquo and

ldquokingrdquo mutually

exclusive Are they

independentKings are

Face cards so they are

NOT mutually exclusive

ldquoSlump I ainrsquot in no slump I just ainrsquot hittinrdquo

Yogi Berra

• PowerPoint Presentation
• Probability
• Law of Large Numbers
• Basic Rules of Probability
• Slide 5
• Slide 6
• Independent
• Slide 8
• Slide 9
• Slide 10
• Slide 11
• Slide 12
• Slide 13
• Slide 14
• Slide 15
• Slide 16
• Slide 17
• Slide 18
• Slide 19
• Slide 20
• Slide 21
• Slide 22
• Slide 23
• Slide 24
• Slide 25
• Slide 26
• Slide 27
• Slide 28
• Slide 29
• Slide 30
• Slide 31
• Slide 32
• Rule 7 Conditional Probability
• Slide 34
• Probabilities from two way tables
• Slide 36
• Slide 37
• Slide 38
• Slide 39
• Slide 40
• Slide 41
• Slide 42
• Slide 43
• Slide 44
• Slide 45
• Slide 46
• Slide 47
• Slide 48

Definition of Independent Events

Two events E and F are independent if and only if

P(F | E) = P(F) or P(E | F) = P(E)

EXAMPLE Illustrating Independent Events

The probability a randomly selected murder victim is male is 07515 The probability a randomly selected murder victim is male given that they are less than 18 years old is 06751

Since P(male) = 07515 and

P(male | lt 18 years old) = 06751

the events ldquomalerdquo and ldquoless than 18 years oldrdquo are not independent In fact knowing the victim is less than 18 years old decreases the probability that the victim is male

I draw one card and look

at it I tell you it is red What is the

probability it is a heart

P( heart | red) =

13P(heart and red) 152

26P(red) 252

Are ldquored cardrdquo and ldquospaderdquo mutually

exclusive Are they

independentA red card canrsquot be a spade so they ARE mutually exclusive

Are ldquored cardrdquo and

ldquoacerdquo mutually

exclusive Are they

independent2 aces are

red cards so they are

NOT mutually exclusive

Are ldquoface cardrdquo and

ldquokingrdquo mutually

exclusive Are they

independentKings are

Face cards so they are

NOT mutually exclusive

ldquoSlump I ainrsquot in no slump I just ainrsquot hittinrdquo

Yogi Berra

• PowerPoint Presentation
• Probability
• Law of Large Numbers
• Basic Rules of Probability
• Slide 5
• Slide 6
• Independent
• Slide 8
• Slide 9
• Slide 10
• Slide 11
• Slide 12
• Slide 13
• Slide 14
• Slide 15
• Slide 16
• Slide 17
• Slide 18
• Slide 19
• Slide 20
• Slide 21
• Slide 22
• Slide 23
• Slide 24
• Slide 25
• Slide 26
• Slide 27
• Slide 28
• Slide 29
• Slide 30
• Slide 31
• Slide 32
• Rule 7 Conditional Probability
• Slide 34
• Probabilities from two way tables
• Slide 36
• Slide 37
• Slide 38
• Slide 39
• Slide 40
• Slide 41
• Slide 42
• Slide 43
• Slide 44
• Slide 45
• Slide 46
• Slide 47
• Slide 48

EXAMPLE Illustrating Independent Events

The probability a randomly selected murder victim is male is 07515 The probability a randomly selected murder victim is male given that they are less than 18 years old is 06751

Since P(male) = 07515 and

P(male | lt 18 years old) = 06751

the events ldquomalerdquo and ldquoless than 18 years oldrdquo are not independent In fact knowing the victim is less than 18 years old decreases the probability that the victim is male

I draw one card and look

at it I tell you it is red What is the

probability it is a heart

P( heart | red) =

13P(heart and red) 152

26P(red) 252

Are ldquored cardrdquo and ldquospaderdquo mutually

exclusive Are they

independentA red card canrsquot be a spade so they ARE mutually exclusive

Are ldquored cardrdquo and

ldquoacerdquo mutually

exclusive Are they

independent2 aces are

red cards so they are

NOT mutually exclusive

Are ldquoface cardrdquo and

ldquokingrdquo mutually

exclusive Are they

independentKings are

Face cards so they are

NOT mutually exclusive

ldquoSlump I ainrsquot in no slump I just ainrsquot hittinrdquo

Yogi Berra

• PowerPoint Presentation
• Probability
• Law of Large Numbers
• Basic Rules of Probability
• Slide 5
• Slide 6
• Independent
• Slide 8
• Slide 9
• Slide 10
• Slide 11
• Slide 12
• Slide 13
• Slide 14
• Slide 15
• Slide 16
• Slide 17
• Slide 18
• Slide 19
• Slide 20
• Slide 21
• Slide 22
• Slide 23
• Slide 24
• Slide 25
• Slide 26
• Slide 27
• Slide 28
• Slide 29
• Slide 30
• Slide 31
• Slide 32
• Rule 7 Conditional Probability
• Slide 34
• Probabilities from two way tables
• Slide 36
• Slide 37
• Slide 38
• Slide 39
• Slide 40
• Slide 41
• Slide 42
• Slide 43
• Slide 44
• Slide 45
• Slide 46
• Slide 47
• Slide 48

I draw one card and look

at it I tell you it is red What is the

probability it is a heart

P( heart | red) =

13P(heart and red) 152

26P(red) 252

Are ldquored cardrdquo and ldquospaderdquo mutually

exclusive Are they

independentA red card canrsquot be a spade so they ARE mutually exclusive

Are ldquored cardrdquo and

ldquoacerdquo mutually

exclusive Are they

independent2 aces are

red cards so they are

NOT mutually exclusive

Are ldquoface cardrdquo and

ldquokingrdquo mutually

exclusive Are they

independentKings are

Face cards so they are

NOT mutually exclusive

ldquoSlump I ainrsquot in no slump I just ainrsquot hittinrdquo

Yogi Berra

• PowerPoint Presentation
• Probability
• Law of Large Numbers
• Basic Rules of Probability
• Slide 5
• Slide 6
• Independent
• Slide 8
• Slide 9
• Slide 10
• Slide 11
• Slide 12
• Slide 13
• Slide 14
• Slide 15
• Slide 16
• Slide 17
• Slide 18
• Slide 19
• Slide 20
• Slide 21
• Slide 22
• Slide 23
• Slide 24
• Slide 25
• Slide 26
• Slide 27
• Slide 28
• Slide 29
• Slide 30
• Slide 31
• Slide 32
• Rule 7 Conditional Probability
• Slide 34
• Probabilities from two way tables
• Slide 36
• Slide 37
• Slide 38
• Slide 39
• Slide 40
• Slide 41
• Slide 42
• Slide 43
• Slide 44
• Slide 45
• Slide 46
• Slide 47
• Slide 48

Are ldquored cardrdquo and ldquospaderdquo mutually

exclusive Are they

independentA red card canrsquot be a spade so they ARE mutually exclusive

Are ldquored cardrdquo and

ldquoacerdquo mutually

exclusive Are they

independent2 aces are

red cards so they are

NOT mutually exclusive

Are ldquoface cardrdquo and

ldquokingrdquo mutually

exclusive Are they

independentKings are

Face cards so they are

NOT mutually exclusive

ldquoSlump I ainrsquot in no slump I just ainrsquot hittinrdquo

Yogi Berra

• PowerPoint Presentation
• Probability
• Law of Large Numbers
• Basic Rules of Probability
• Slide 5
• Slide 6
• Independent
• Slide 8
• Slide 9
• Slide 10
• Slide 11
• Slide 12
• Slide 13
• Slide 14
• Slide 15
• Slide 16
• Slide 17
• Slide 18
• Slide 19
• Slide 20
• Slide 21
• Slide 22
• Slide 23
• Slide 24
• Slide 25
• Slide 26
• Slide 27
• Slide 28
• Slide 29
• Slide 30
• Slide 31
• Slide 32
• Rule 7 Conditional Probability
• Slide 34
• Probabilities from two way tables
• Slide 36
• Slide 37
• Slide 38
• Slide 39
• Slide 40
• Slide 41
• Slide 42
• Slide 43
• Slide 44
• Slide 45
• Slide 46
• Slide 47
• Slide 48

Are ldquored cardrdquo and

ldquoacerdquo mutually

exclusive Are they

independent2 aces are

red cards so they are

NOT mutually exclusive

Are ldquoface cardrdquo and

ldquokingrdquo mutually

exclusive Are they

independentKings are

Face cards so they are

NOT mutually exclusive

ldquoSlump I ainrsquot in no slump I just ainrsquot hittinrdquo

Yogi Berra

• PowerPoint Presentation
• Probability
• Law of Large Numbers
• Basic Rules of Probability
• Slide 5
• Slide 6
• Independent
• Slide 8
• Slide 9
• Slide 10
• Slide 11
• Slide 12
• Slide 13
• Slide 14
• Slide 15
• Slide 16
• Slide 17
• Slide 18
• Slide 19
• Slide 20
• Slide 21
• Slide 22
• Slide 23
• Slide 24
• Slide 25
• Slide 26
• Slide 27
• Slide 28
• Slide 29
• Slide 30
• Slide 31
• Slide 32
• Rule 7 Conditional Probability
• Slide 34
• Probabilities from two way tables
• Slide 36
• Slide 37
• Slide 38
• Slide 39
• Slide 40
• Slide 41
• Slide 42
• Slide 43
• Slide 44
• Slide 45
• Slide 46
• Slide 47
• Slide 48

Are ldquoface cardrdquo and

ldquokingrdquo mutually

exclusive Are they

independentKings are

Face cards so they are

NOT mutually exclusive

ldquoSlump I ainrsquot in no slump I just ainrsquot hittinrdquo

Yogi Berra

• PowerPoint Presentation
• Probability
• Law of Large Numbers
• Basic Rules of Probability
• Slide 5
• Slide 6
• Independent
• Slide 8
• Slide 9
• Slide 10
• Slide 11
• Slide 12
• Slide 13
• Slide 14
• Slide 15
• Slide 16
• Slide 17
• Slide 18
• Slide 19
• Slide 20
• Slide 21
• Slide 22
• Slide 23
• Slide 24
• Slide 25
• Slide 26
• Slide 27
• Slide 28
• Slide 29
• Slide 30
• Slide 31
• Slide 32
• Rule 7 Conditional Probability
• Slide 34
• Probabilities from two way tables
• Slide 36
• Slide 37
• Slide 38
• Slide 39
• Slide 40
• Slide 41
• Slide 42
• Slide 43
• Slide 44
• Slide 45
• Slide 46
• Slide 47
• Slide 48

ldquoSlump I ainrsquot in no slump I just ainrsquot hittinrdquo

Yogi Berra

• PowerPoint Presentation
• Probability
• Law of Large Numbers
• Basic Rules of Probability
• Slide 5
• Slide 6
• Independent
• Slide 8
• Slide 9
• Slide 10
• Slide 11
• Slide 12
• Slide 13
• Slide 14
• Slide 15
• Slide 16
• Slide 17
• Slide 18
• Slide 19
• Slide 20
• Slide 21
• Slide 22
• Slide 23
• Slide 24
• Slide 25
• Slide 26
• Slide 27
• Slide 28
• Slide 29
• Slide 30
• Slide 31
• Slide 32
• Rule 7 Conditional Probability
• Slide 34
• Probabilities from two way tables
• Slide 36
• Slide 37
• Slide 38
• Slide 39
• Slide 40
• Slide 41
• Slide 42
• Slide 43
• Slide 44
• Slide 45
• Slide 46
• Slide 47
• Slide 48