MTH120_Chapter5

Copyright © 2002 by The McGraw-Hill Companies, Inc. All rights reserved. McGraw-Hill/Irwin

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Chapter FiveA Survey of Probability A Survey of Probability ConceptsConceptsGOALS

When 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.

FOURDefine the terms: conditional probability and joint probability.


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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:

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

SIXUse a tree diagram to organize and compute probabilities.

SEVEN Calculate a probability using Bayes’ theorem.


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DefinitionsDefinitions

A probability is a measure of the likelihood that an event in the future will happen.

It it can only assume a value between 0 and 1.

A value near zero means the event is not likely to happen. A value near one means it is likely.

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


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DefinitionsDefinitions continuedcontinued

The classical definition applies when there are n equally likely outcomes.

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

Subjective probability is based on whatever information is available.


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Definitions Definitions continuedcontinued

An experiment is the observation of some activity or the act of taking some measurement.

An outcome is the particular result of an experiment.

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


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Mutually Exclusive EventsMutually Exclusive Events

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

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


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Collectively Exhaustive EventsCollectively Exhaustive Events

Events are collectively exhaustive if at least one of the events must occur when an experiment is conducted.


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A fair die is rolled once. The experiment is rolling the die.The possible outcomes are the numbers 1, 2,

3, 4, 5, and 6. An event is the occurrence of an even

number. That is, we collect the outcomes 2, 4, and 6.

Example 1Example 1


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EXAMPLE 2EXAMPLE 2

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:

155.01200

186)( AP


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Subjective ProbabilitySubjective 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 ProbabilityBasic Rules of Probability

If two events A and B are mutually exclusive, the special rule of addition states that the probability of A or B occurring equals the sum of their respective probabilities:

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


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EXAMPLE 3EXAMPLE 3

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

Arrival Frequency

Early 100

On Time 800

Late 75

Canceled 25

Total 1000


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EXAMPLE 3 EXAMPLE 3 continuedcontinued

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.

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

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


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The Complement RuleThe Complement Rule

The complement rule is used to determine the probability of an event occurring by subtracting the probability of the event not occurring from 1.

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).


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The Complement Rule The Complement Rule continuedcontinued

A Venn diagram illustrating the complement rule would appear as:

A ~A


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EXAMPLE 4EXAMPLE 4

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.

If D is the event that a flight is canceled, then P(D) = 25/1000 = .025.


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P(A or B) = 1 - P(C or D)

= 1 - [.8 +.025] =.175

C.8

D.025

~(C or D) = (A or B) .175


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The General Rule of AdditionThe 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 AdditionThe General Rule of Addition

The Venn Diagram illustrates this rule:

A and B

A

B


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EXAMPLE 5EXAMPLE 5

In a sample of 500 students, 320 said they had a stereo, 175 said they had a TV, and 100 said they had both:

Stereo 320

Both 100

TV175


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

P(S) = 320/500 = .64.

P(T) = 175/500 = .35.

P(S and T) = 100/500 = .20.


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If a student is selected at random, what is the probability that the student has either a stereo or a TV in his or her room?

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

= .64 +.35 - .20 = .79.


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Joint ProbabilityJoint Probability

A joint 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 MultiplicationSpecial Rule of Multiplication

The special rule of multiplication requires that two events A and B are independent.

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)


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EXAMPLE 6EXAMPLE 6

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?

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


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What is the probability that at least one of these stocks increase in value during the next year? (This means that either one can increase or both.)

P(at least one) = (.5)(.3) + (.5)(.7) +(.7)(.5)

= .85.


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Conditional ProbabilityConditional Probability

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

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


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General Multiplication RuleGeneral Multiplication Rule

The general rule of multiplication is used to find the joint probability that two events will occur.

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.


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General Multiplication RuleGeneral 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 7EXAMPLE 7

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

MAJOR Male Female Total

Accounting 170 110 280

Finance 120 100 220

Marketing 160 70 230

Management 150 120 270

Total 600 400 1000


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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?

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

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


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Tree DiagramsTree Diagrams

A tree diagram is useful for portraying conditional and joint probabilities. It is particularly useful for analyzing business decisions involving several stages.

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.


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R1

B1

R2

B2

R2

B2

7/12

5/12

6/11

5/11

7/11

4/11


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Bayes’ TheoremBayes’ Theorem

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

It is computed using the following formula:

)/()()/()(

)/()()|(

2211

111 ABPAPABPAP

ABPAPBAP


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EXAMPLE 9EXAMPLE 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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The following table summarizes the Duff production experience.

% of TotalProduction

% of under-filled bottles

A 55 3

B 45 4


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Example 9 Example 9 continuedcontinued

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 CountingSome Principles of Counting

The multiplication formula 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.

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

(10)(8) = 80


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

Note: The order of arrangement is important in permutations.

)!(

!

rn

nP rn


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

)!(!

!

rnr

nCrn


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EXAMPLE 11EXAMPLE 11

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?

792)!512(!5

!12512

C


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Example 11 Example 11 continuedcontinued

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

040,95)!512(

!12512

P

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MTH120_Chapter5