chapter 3: probability statistics. mcclave: statistics, 11th ed. chapter 3: probability 2 where...

Chapter 3: Probability

Statistics

McClave: Statistics, 11th ed. Chapter 3: Probability

2

Where We’ve Been

Making Inferences about a Population Based on a Sample

Graphical and Numerical Descriptive Measures for Qualitative and Quantitative Data


3

Where We’re Going

Probability as a Measure of Uncertainty

Basic Rules for Finding Probabilities Probability as a Measure of Reliability

for an Inference


4

3.1: Events, Sample Spaces and Probability

An experiment is an act or process of observation that leads to a single outcome that cannot be predicted with certainty.


5


A sample point is the most basic outcome of an experiment.

A four

A Head

An Ace


6


A sample space of an experiment is the collection of all sample points. Roll a single die:

S: {1, 2, 3, 4, 5, 6}


7


Sample points and sample spaces are often represented with Venn diagrams.


8


All probabilities must be between 0 and 1.

The probabilities of all the sample points must sum to 1.

Probability Rules for Sample Points

1

1i

n

i

p

0 1ip


9


All probabilities must be between 0 and 1

The probabilities of all the sample points must sum to 1

Probability Rules for Sample Points

1

1i

n

i

p

0 1ip

0 indicates an impossible outcome and 1 a certain outcome.

Something has to happen.


10


An event is a specific collection of sample points: Event A: Observe an even number.


11

3.1: Events, Sample Spaces and Probability The probability of an event is the sum of the

probabilities of the sample points in the sample space for the event. Event A: Observe an even number. P(A) = 1/6 + 1/6 + 1/6 = 3/6 = 1/2


12


Calculating Probabilities for Events Define the experiment. List the sample points. Assign probabilities to sample points. Collect all sample points in the event of

interest. The sum of the sample point probabilities is

the event probability.


13


Calculating Probabilities for Events Define the experiment List the sample points Assign probabilities to sample points Collect all sample points in the event of

interest The sum of the sample point probabilities is

the event probability

If the number of sample points gets too large, we need a way to keep track of how they can be combined for different events.


14


If a sample of n elements is drawn from a set of N elements (N ≥ n), the number of different samples is

!

.! )!

N N

n n N n


15


If a sample of 5 elements is drawn from a set of 20 elements, the number of different samples is

20 20! 20 19 18 3 2 1

15,504.5 5! 20 5)! 5 4 3 2 1 15 14 13 3 2 1


16

3.2: Unions and Intersections

Compound EventsMade of two or

more other events

UnionA B

Either A or B, or both, occur

IntersectionA B

Both A and Boccur


17



18


A B

B C

A

B C

A C

A B C


19

3.3: Complementary Events

The complement of any event A is the event that A does not occur, AC.

A: {Toss an even number}AC: {Toss an odd number}

B: {Toss a number ≤ 3}BC: {Toss a number ≥ 4}

A B = {1,2,3,4,6}[A B]C = {5}(Neither A nor B occur)


20

3.3: Complementary Events

)(1)(

)(1)(

1)()(

APAP

APAP

APAP

C

C

C


21

3.3: Complementary EventsA: {At least one head on two coin flips}

AC: {No heads}

43

41

41

43

41

41

41

1)(1)(

)(

)(

}{:

},,{:

C

C

C

APAP

AP

AP

TTA

THHTHHA


22

3.4: The Additive Rule and Mutually Exclusive Events

The probability of the union of events A and B is the sum of the probabilities of A and B minus the probability of the intersection of A and B:

)()()()( BAPBPAPBAP


23


P(Surgery or Obstetrics) ( )

( ) ( ) ( ) ( )

( ) .12 .16 .02 .26

P A B

P A B P A P B P A B

P A B

At a particular hospital, the probability of a patient having surgery (Event A) is .12, of an obstetric treatment (Event B) .16, and of both .02.

What is the probability that a patient will have either treatment?


24


Events A and B are mutually exclusive if A B contains no sample points.


25


)()()( BPAPBAP

If A and B are mutually exclusive,

3.5: Conditional Probability

Additional information or other events occurring may have an impact on the probability of an event.

26McClave: Statistics, 11th ed. Chapter 3: Probability


Additional information may have an impact on the probability of an event. P(Rolling a 6) is one-sixth

(unconditionally). If we know an even number was

rolled, the probability of a 6 goes up to one-third.



The sample space is reduced to only the conditioning event.

To find P(A), once we know B has occurred (i.e., given B), we ignore BC (including the A region within BC).


B BC

A A



B BC

A A

)(

)()(

BP

BAPBAP



B

A

)(

)()(

BP

BAPBAP


55% of sampled executives had cheated at golf (event A). P(A) = .55

20% of sampled executives had cheated at golf and lied in business (event B). P(A B) = .20

What is the probability that an executive had lied in business, given s/he had cheated in golf, P(B|A)?



P(A) = .55

P(A B) = .20 What is P(B|A)? )(

)()(

AP

ABPABP

364.55.

20.)( ABP


3.6: The Multiplicative Rule and Independent Events

The conditional probability formula can be rearranged into the Multiplicative Rule of Probability to find joint probability.

)(

)()(

BP

BAPBAP

)()()(

)()()(

ABPAPBAP

or

BAPBPBAP



Assume three of ten workers give illegal deductions Event A: {First worker selected gives an illegal deduction} Event B: {Second worker selected gives an illegal

deduction} P(A) = P(B) = .1 + .1 + .1 = .3

P(B|A) has only nine sample points, and two targeted workers, since we selected one of the targeted workers in the first round: P(B|A) = 1/9 + 1/9 = .11 + .11 = .22

The probability that both of the first two workers selected will have given illegal deductions P(AB) = P(B|A)P(A) = .(3) (.22) = .066



35



A Tree Diagram

First selected worker

Second selected worker



Dependent Events P(A|B) ≠ P(A) P(B|A) ≠ P(B)

Independent Events P(A|B) = P(A) P(B|A) = P(B)

Studying stats 40 hours per week

Working 40 hours per week

Having blue eyes


3.7: Random Sampling

If n elements are selected from a population in such a way that every set of n elements in the population has an equal probability of being selected, the n elements are said to be a (simple) random sample.



How many five-card poker hands can be dealt from a standard 52-card deck?

Use the combination rule:

960,598,2)!552(!5

!52

5

52

n

N



Random samples can be generated by Mixing up the elements and drawing by hand,

say, out of a hat (for small populations) Random number generators Random number tables

Table I, App. B Random sample/number commands on

software


3.8: Some Additional Counting Rules

Situation

Multiplicative Rule Draw one element from each

of k sets, sized n1, n2, n3, … nk

Permutations Rule Draw n elements, arranged

in a distinct order, from a set of N elements

Partitions Rule Partition N elements into k

groups, sized n1, n2, n3, … nk (ni=N)

Number of Different Results


42

!!!!

!

)!(

!

321

321

k

Nn

k

nnnn

N

nN

NP

nnnn


Multiplicative RuleAssume one professor from each of the six departments in a division will be selected for a special committee. The various departments have four, seven, six, eight, six and five professors eligible. How many different committees, X*, could be formed?


43

360,87

568674

* 321

knnnnX


Permutations If there are eight horses entered in a race, how many different win, place and show possibilities are there?


44

336)!38(

!8

)!(

!

nN

NPNn

3.8: Some Additional Counting Rules Partitions Rule

The technical director of a theatre has twenty stagehands. She needs eight electricians, ten carpenters and two props people. How many different allocations of stagehands, Y*, can there be?


45

1 2 3

! 20!* 8,314,020

! ! ! ! 8!10!2!k

NY

n n n n

3.8: Bayes’s Rule

Given k mutually exclusive and exhaustive events B1, B2,… Bk, and an observed event A, then

)

1 1 2 2

( | ) ( ) / ( )

( ) ( |.

( ) ( | ) ( ) ( | ) ( ) ( | )

i i

i i

k k

P B A P B A P A

P B P A B

P B P A B P B P A B P B P A B


3.8: Bayes’s Rule

Suppose the events B1, B2, and B3, are mutually exclusive and complementary events with P(B1) = .2, P(B2) = .15 and P(B3) = .65. Another event A has these conditional probabilities: P(A|B1) = .4, P(A|B2) = .25 and P(A|B3) = .6.

What is P(B1|A)?

158.5075.

08.

39.0375.08.

08.

6.65.25.15.4.2.

4.2.

)|()()|()()|()(

)|()(

)(/)()|(

332211

11

11

BAPBPBAPBPBAPBP

BAPBP

APABPABP


chapter 3: probability statistics. mcclave: statistics, 11th ed. chapter 3: probability 2 where...

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