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Chapter

Five

McGraw-Hill/Irwin

© 2005 The McGraw-Hill Companies, Inc., All Rights Reserved.

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Chapter FiveA Survey of Probability A Survey of Probability

ConceptsConceptsGOALSWhen you have completed this chapter, you will be able to:

ONEDefine probability.

TWO Describe the classical, empirical, and subjective approaches to probability.

THREEUnderstand the terms: experiment, event, outcome, permutations, and combinations.

Goals

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Chapter Five continuedA Survey of Probability A Survey of Probability ConceptsConceptsGOALS

When you have completed this chapter, you will be able to:FOURDefine the terms: conditional probability and joint probability.

FIVE Calculate probabilities applying the rules of addition and the rules of multiplication.

SIXUse a tree diagram to organize and compute probabilities.

Goals

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SEVEN Calculate a probability using Bayes’ theorem.

A Survey of Probability A Survey of Probability ConceptsConcepts

Chapter Five continued

Goals

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Movie

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Definitions continued

There are three definitions of probability: classical, empirical, and subjective.

The

ClassicalClassical definition

applies when there are n

equally likely outcomes.

The EmpiricalEmpirical definition applies when the number of times the event happens is divided by the number of observations.

SubjectiveSubjective probability is based on whatever information is available.

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Movie

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Definitions continued

An EventEvent is the collection of one or more outcomes of an experiment.

An OutcomeOutcome is the particular result of an experiment.

Experiment:Experiment: A fair die is cast. A fair die is cast.

Possible outcomes: The Possible outcomes: The numbers 1, 2, 3, 4, 5, 6 numbers 1, 2, 3, 4, 5, 6

One possible event: The One possible event: The occurrence of an even occurrence of an even number. That is, we collect number. That is, we collect the outcomes 2, 4, and 6.the outcomes 2, 4, and 6.

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Mutually Exclusive Events

Events are IndependentIndependent if the occurrence of one event does not affect the occurrence of another.

Events are Mutually Mutually ExclusiveExclusive if the occurrence of any one event means that none of the others can occur at the same time.

Mutually exclusive: Mutually exclusive: Rolling a 2 precludes Rolling a 2 precludes rolling a 1, 3, 4, 5, 6 rolling a 1, 3, 4, 5, 6 on the same roll.on the same roll.

Independence: Rolling a 2 on the first throw does not influence the probability of a 3 on the next throw. It is still a one in 6 chance.

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Collectively Exhaustive Events

Events are Collectively ExhaustiveCollectively Exhaustive if at least one of the events must occur when an experiment is conducted.

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Example 2

155.01200

186)( AP

Throughout her teaching career Professor Jones has awarded 186 A’s out of 1,200 students. What is the probability that a student in her section this semester will receive an A?

This is an example of the empirical definition of probability.

To find the probability a selected student earned an A:

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Subjective Probability

Examples of subjective probability are:

estimating the probability the Washington Redskins will win the Super Bowl this year.

estimating the probability mortgage rates for home loans will top 8 percent.

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Basic Rules of ProbabilityP(A or B) = P(A) + P(B)

If two events A and B are mutually

exclusive, the

Special Rule of Special Rule of AdditionAddition states that the

Probability of A or B occurring equals the sum of

their respective probabilities.

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Example 3

Arrival Frequency

Early 100

On Time 800

Late 75

Canceled 25

Total 1000

New England Commuter Airways recently supplied the following information on their commuter flights from Boston to New York:

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Example 3 continued

The probability that a flight is either early or late is:

P(A or B) = P(A) + P(B) = .10 + .075 =.175.

If A is the event that a flight arrives early, then P(A) = 100/1000 = .10.

If B is the event that a flight arrives late, then P(B) = 75/1000 = .075.

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The Complement Rule

If P(A) is the probability of event A and P(~A) is the complement of A,

P(A) + P(~A) = 1 or P(A) = 1 - P(~A).

The Complement RuleComplement Rule is used to determine the probability of an event occurring by subtracting the probability of the event not occurring from 1.

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The Complement Rule continued

A~~AA

A Venn DiagramVenn Diagram illustrating the complement rule would appear as:

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Example 4

If D is the event that a flight is canceled, then

P(D) = 25/1000 = .025.

Recall example 3. Use the complement rule to find the probability of an early (A) or a late (B) flight

If C is the event that a flight arrives on time, then

P(C) = 800/1000 = .8.

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Example 4 continued

CC.8.8

DD.025.025

~(C or D) = (A or B)~(C or D) = (A or B) .175.175

P(A or B) = 1 - P(C or D) = 1 - [.8 +.025] =.175

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The General Rule of Addition

If A and B are two events that are not mutually exclusive, then P(A or B) is given by the following formula:

P(A or B) = P(A) + P(B) - P(A and B)

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The General Rule of Addition

A and BA and B

AA

BB

The Venn Diagram illustrates this rule:

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EXAMPLE 5

StereoStereo 320320

BothBoth 100100

TVTV175175

In a sample of 500 students, 320 said they had a stereo, 175 said they had a TV, and 100 said they had both. 5 said they had neither.

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Example 5 continued

P(S or TV) = P(S) + P(TV) - P(S and TV)

= 320/500 + 175/500 – 100/500

= .79.

P(S and TV) = 100/500

= .20

If a student is selected at random, what is the probability that the student has only a stereo or TV? What is the probability that the student has both a stereo and TV?

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Joint Probability

A Joint ProbabilityJoint Probability measures the likelihood that two or more events will happen concurrently.

An example would be the event that a student has both a stereo and TV in his or her dorm room.

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Special Rule of Multiplication

Two events A and B are independent if the occurrence of one has no effect on the probability of the occurrence of the other.

This rule is written: P(A and B) = P(A)P(B)

The Special Rule of MultiplicationSpecial Rule of Multiplication requires that two events A and B are

independent.

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Example 6

5-year stock prices

0

5

10

15

20

25

30

35

40

45

1 2 3 4 5

Year

Stoc

k pr

ice

$

IBM

GE

P(IBM and GE) = (.5)(.7) = .35

Chris owns two stocks, IBM and General Electric (GE). The probability that IBM stock will increase in value next year is .5 and the probability that GE stock will increase in value next year is .7. Assume the two stocks are independent. What is the probability that both stocks will increase in value next year?

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Example 6 continued

P(at least one)

= P(IBM but not GE)

+ P(GE but not IBM)

+ P(IBM and GE)

W h at is th e

p ro b a b ility th a t a t lea st

o n e o f th ese sto ck s in creases in v a lu e in

th e n ex t y ear? T h is m ea n s th a t

e ith er o n e ca n in crease or

b o th .

(.5)(1-.7) + (.7)(1-.5) + (.7)(.5) = .85

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Conditional Probability

The probability of event A occurring given that the event B has occurred is written P(A|B).

A Conditional ProbabilityConditional Probability is the probability of a particular event occurring, given that another event has occurred.

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General Multiplication Rule

It states that for two events A and B, the joint probability that both events will happen is found by multiplying the probability that event A will happen by the conditional probability of B given that A has occurred.

The General General Rule of Rule of

MultiplicationMultiplication is used to find the joint probability that two events will occur.

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General Multiplication Rule

The joint probability, P(A and B), is given by the

following formula:

P(A and B) = P(A)P(B/A)

or P(A and B) =

P(B)P(A/B)

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Example 7

Major Male Female Total

Accounting 170 110 280

Finance 120 100 220

Marketing 160 70 230

Management 150 120 270

Total 600 400 1000

The Dean of the School of Business at Owens University collected the following information about undergraduate students in her college:

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Example 7 continued

P(A|F) = P(A and F)/P(F)

= [110/1000]/[400/1000] = .275

If a student is selected at random, what is the probability that the student is a female (F) accounting major (A)?

P(A and F) = 110/1000.

Given that the student is a female, what is the probability that she is an accounting major?

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Tree Diagrams

Example 8: In a bag containing 7 red chips and 5 blue chips you select 2 chips one after the other without replacement. Construct a tree diagram showing this information.

A Tree DiagramTree Diagram is useful for portraying conditional and joint probabilities. It is particularly useful for analyzing business decisions involving several stages.

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Example 8 continued

R1

B1

R2

B2

R2

B2

7/12

5/12

6/11

5/11

7/11

4/11

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Bayes’ Theorem

)/()()/()(

)/()()|(

2211

111 ABPAPABPAP

ABPAPBAP

Bayes’ Theorem is a method for revising a probability given additional information.

It is computed using the following formula:

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Example 9

Duff Cola Company recently received several complaints that their bottles are under-filled. A complaint was received today but the production manager is unable to identify which of the two Springfield plants (A or B) filled this bottle. What is the probability that the under-filled bottle came from plant A?

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Example 9 continued

The following table summarizes the Duff production experience.

 % of total production

% of underfilled

bottle

A 5.5 3.0

B 4.5 4.0

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Example 9 continued

4783.)04(.45.)03(.55.

)03(.55.

)/()()/()(

)/()()/(

BUPBPAUPAP

AUPAPUAP

The likelihood the bottle was filled in Plant A is reduced from .55 to .4783.

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Some Principles of Counting

Example 10: Dr. Delong has 10 shirts and 8 ties. How many shirt and tie outfits does he have?

The Multiplication Multiplication FormulaFormula indicates that if there are m ways of doing one thing and n ways of doing another thing, there are m x n ways of doing both.

(10)(8) = 80

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Some Principles of Counting

)!(

!

rn

nP rn

A PermutationPermutation is any arrangement of r objects selected from n possible objects.

Note: The order of arrangement is important in permutations.

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Some Principles of Counting

)!(!

!

rnr

nCrn

A CombinationCombination is the number of ways to choose r objects from a group of n objects without regard to order.

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Example 11

792)!512(!5

!12512

C

There are 12 players on the Carolina Forest High School basketball team. Coach Thompson must pick five players among the twelve on the team to comprise the starting lineup. How many different groups are possible? (Order does not matter.)

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Example 11 continued

040,95)!512(

!12512

P

Suppose that in addition to selecting the group, he must also rank each of the players in that starting lineup according to their ability (order matters).