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PROBABILITY Probability Concepts

Theoretical Probability vs Relative

Frequency

Calculating Probabilities

Venn Diagrams

Intersection of Sets

Union of Sets

Mutually Exclusive Events

Inclusive Events

Complementary Events

Probability Games in Life 1

Example: Tossing a coin twice

Probability

The likelihood or chance of an event happening

E.g. Heads (50%) or Tails (50%)

Outcomes

The possible results of an experiment

E.g. Heads; Tails / Heads; Heads /

Tails ; Heads / Tails; Tails

Sample Space

The set of all possible outcomes: S={HH,HT,TH,TT}

The number of outcomes: (S) = 4

The Coin Toss

PROBABILITY CONCEPTS

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Types of Events

a) Certain events

These events will always happen.

E.g. If you toss a coin twice and the event is

E = {getting a head or a tail}

then you are certain to get a head or tail on each toss.

(b) Even chance events

50% probability of an event happening

E.g. Consider the experiment with S = {HH, HT, TH, TT} and the event A = {getting two heads or two tails)

then event A has 50% chance of happening = {HH, TT}

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c) Equally likely events

Each event has an equally likely chance of happening.

Events are unbiased

E.g. If you toss a coin, there is 50% change of getting

a Head and 50% change of a Tail

(d) Impossible Events

The event cannot happen

E.g. G = {getting the number 7 from a die that is thrown}. This is impossible because the die doesn't contain the number 7. The chance of this happening is= 0.

G= or G = { }, which is called the empty set.

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The probability scale

We write probabilities as fractions, decimals or

percentages.

The less likely an event is to happen, the

smaller the fraction.

The more likely the probability, the greater the

fraction.

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THEORETICAL PROBABILITY VS

RELATIVE FREQUENCY

Theoretical Probability

When all outcomes of an activity are equally likely,

you can calculate the probability of an event

happening by using the following definition:

P(E) = number of favorable outcomes = n (E)

total number of possible outcomes = n (S)

Probability of throwing a dice and getting a 4 is

P(4) = 1

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The Probability of an Event Marble Example 6

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Relative Frequency

This method involves many trials.

E.g. Suppose you threw a die 200 times and then

counted the number of times you got the number 4.

You might land up getting this number 33 times. You

would then say that the relative frequency of getting

the number 4 is = = 14.5%.

Only after many trials will the relative frequency get

closer to the theoretical probability of . = 16.67%.

200

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The Spinner

Basic Probability Example

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CALCULATING PROBABILITIES

Example 1

Suppose that you are required to draw cards

numbered from 1 to 12 out of a hat. The sample

space is S = {1,2,3,4,5,6,7,8,9,10,11,12}

Consider the following events:

a) Event A = {drawing prime numbers} = {2,3,5,7,11}

b) Event B= {drawing odd numbers} ={1,3,5,7,9,11}

12

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Example 2

The sample space is S = {1,2,3,4,5,6,7,8,9,10,11,12}

c) Event C = {drawing factors of 6} = {1,2,3,6}

d) Event D = {drawing numbers greater than 12} = { }

e) Event E= {drawing natural number} = {1,2,3,4,5,6,7,8,9,10,11,12}

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Example 3

Consider the word MATHEMATICS. A letter is chosen from this word.

(a) Find the probability that the letter chosen is M.

There are a total of 11 letters in the word. M appears twice. Therefore the probability of getting M is .

(b) Find the probability that the letter chosen is not M.

The probability is

(c) Find the probability that the letter chosen is a

vowel {A,E,I,0,U}.

The probability is

11

2

11

9

11

4

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EXERCISE

1. 300 tickets were sold for a raffle. Jason bought 12

tickets. What is the probability that he:

(a) will win the prize?

(b) will not win the prize?

2. There are 60 boys and 40 girls auditioning for POP

IDOLS. The names of the contestants are put into a

hat and a name is drawn. What is the probability that:

(a) a boy will be selected on the first draw?

(b) a girl will be selected on the first draw?

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3. A bag contains 6 blue marbles, 5 red marbles,

8 green marbles and 9 white marbles. What is the

probability of:

(a) drawing a white marble?

(b) drawing a green marble?

(c) drawing a blue marble?

(d) drawing a red marble?

(e) drawing a red or blue marble?

(f) drawing a blue or green marble?

(g) drawing a pink marble?

(h) drawing a white, green or red marble?

(i) not drawing a blue marble?

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4. A card is drawn from a pack of 52 cards.

Determine the probability of drawing:

(a) a heart.

(b) a jack of clubs

(c) an ace.

(d) a king or queen

(e) neither a heart or a spade.

5. A six-sided die is thrown. Determine the probability of:

(a) throwing a 6.

(b) throwing a 2 or a 4.

(c) throwing an odd number.

(d) not throwing a 5.

Probability Quiz 13

VENN DIAGRAMS

Venn diagrams represent the sample space.

Example

Consider the hat experiment where

S = {1,2,3,4,5,6,7,8,9,10,11,12}.

Suppose that there are two events:

A = {drawing numbers less than or equal to 6}

= {1,2,3,4,5,6}

B = {drawing numbers greater than 6}

= {7,8,9,10,11,12}

Represent this on a Venn Diagram:

Basic Venn Diagrams

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INTERSECTION OF SETS

The intersection occurs where the elements share a

common space. These are called inclusive events.

Example

Consider the hat experiment

where S = {l, 2,3,4,5,6,7,8,9,10,11,12}.

There are 2 events:

C = {drawing a factor of 6} = {1,2,3,6}

D = {drawing a factor of 9} = {l, 3,9}

Find the elements that intersect.

C D={l,3}

C and D ={1,3}

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UNION OF SETS

The union of A and B is an event consisting of all outcomes

that are in A or B.

Example

Determine the union of C and D.

C D ={l,2,3,6,9}

C or D ={1,2,3,6,9}

Here the numbers 4,5,7,8,10,11,12 are excluded from the

union of C and D. The number 1 and 3 appear in both set

C and D and are written only once in the union set.

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MUTUALLY EXCLUSIVE EVENTS

Events with no elements in common. Event A and B

exclude each other. If A happens, then B cannot happen.

Both cannot happen at the same time.

Example

a) Find the intersection of A and B:

A B = { }

A and B = { } empty set

P(A B) = 0

b) Find the union of A and B:

A B = {1;2;3;4;5;6;7;8;9;10;11;12}

A or B = {1;2;3;4;5;6;7;8;9;10;11;12}

P(A B) = P(A) + P(B) – P( A B)

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INCLUSIVE EVENTS

Events with elements in common.

Example

a) Find the intersection of C and D:

C D = {1;3}

C and D = {1;3 }

b) Find the union of C and D:

C D = {1;2;3;6;9}

C or D = {1;2;3;6;9}

P(C D) = P(A) + P(B) - P(A B)

Practicing Venn

Diagrams

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COMPLEMENTARY EVENTS

If Event A and Event B is mutually exclusive, then Event A

and Event B are complementary.

Example

a) Find the complement of A.

A = {1;2;3;4;5;6}

Complement of A = Not A (A’)

= B = {7;8;9;10;11;12}

P(A) + P(A')= 1

b) Find the complement of B.

B = {7;8;9;10;11;12}

Complement of B = Not B (B’)

= A = {1;2;3;4;5;6}

P(B) + P(B')= 1 …. P (not B) = 1 - P(B)

Playing Cards &

Venn Diagrams 19

EXERCISE

Cards numbered from 1 to 12 are put into a box and shaken. Cards are then drawn and replaced. The following events are given:

A = {drawing an even number}

B = {drawing an odd number}

C = {drawing a number greater than 7}

D = {drawing a number less than 5}

E = {drawing natural numbers less than 7}

F = {drawing natural numbers greater than 4}

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(a) Draw a Venn Diagram to represent events A and B.

(b) Determine P(A B)

(c) Determine P(A B)

(d) Show that A and B are mutually exclusive.

(e) Are events A and B complementary?

(f) Draw a Venn Diagram to represent events A and C.

(g) Determine P(A or C)

(h) Determine P(A and C)

(i) Show that A and C are inclusive.

(j) Are events A and C complementary?

(k) Draw a Venn Diagram to represent events C and D.

(1) Determine whether C and D are mutually exclusive or

inclusive; complementary or not complementary.

(m) Draw a Venn Diagram to represent events E and F.

(n) Determine whether E and F are mutually exclusive or

inclusive; complementary or not complementary.

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Conditional Probability: Pick the Correct Door!

Picking Cards or Rolling Die

PROBABILITY GAMES IN LIFE

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