Chapter 05 A Survey of Probability Concepts

5/20/2018 Chapter 05 A Survey of Probability Concepts

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Chapter

Five

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


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Chapter Five

A Survey of Probability Concepts

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

ONE

Define probability.TWO

Describe the classical, empirical, and subjective approaches

to probability.

THREE

Understand the terms: experiment, event, outcome,

permutations, and combinations.Goals


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Chapter Five continued


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

FOUR

Define the terms: conditional probability and jointprobability.

FIVE

Calculate probabilities applying the rules of addition and the

rules of multiplication.

SIX

Use a tree diagram to organize and compute probabilities.

Goals


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SEVEN

Calculate a probability using Bayes theorem.


Chapter Five continued

Goals


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Movie


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Definitionscontinued

There are three definitions of probability: classical,

empirical, and subjective.

The

Classical

definitionapplies when

there are n

equally likely

outcomes.

The Empiricaldefinition applies

when the numberof times the event

happens is

divided by the

number ofobservations.

Subjectiveprobability is

based on

whatever

information is

available.


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Movie


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Definitions continued

An Event isthe collection

of one or moreoutcomes of an

e x p e r i m e n t .

An Outcome is

t he pa r t i cu l a rr e s u l t o f a n

e x p e r i m e n t .

Experiment: A fair die is cast.

Possible outcomes: Thenumbers 1, 2, 3, 4, 5, 6

One possible event: The

occurrence of an even

number. That is, we collect

the outcomes 2, 4, and 6.


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Mutually Exclusive Events

Events are Independentif the occurrence of one event

does not affect the occurrence

of another.

Events are Mutually

Exclusiveif theoccurrence of any one

event means that none

of the others can occurat the same time.

Mutually exclusive:

Rolling a 2 precludes

rolling a 1, 3, 4, 5, 6

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 Exhaustiveif at least one of the events must occur

when an experiment is conducted.


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Example 2

155.01200186)( AP

Throughou t he r

t e a c h i n g c a re e r

Professor Jones has

awarded 186 Asout

of 1,200 students.

W h a t i s t h eprobability that a

s t u d e n t i n h e r

s e c t i o n t h i s

s e m e s t e r w i l lr e c e i v e a n 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 th is year .

est imating the probabil i ty

mortgage rates for home loans

w i l l t o p 8 p e r c e n t .


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

If two events

A and B are mutually

exclusive, the

Special Rule of

Additionstates that the

Probability of A or Boccurring equals the sum of

their respective

probabilities.


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Example 3

Arrival Frequency

Early (A) 100On Time (C) 800

Late (B) 75

Canceled (D) 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(AorB) = P(A) + P(B) = .10 + .075 =.175.

IfAis the event that a

flight arrives early,

thenP(A) = 100/1000

= .10.

IfBis the event that a

flight arrives late, thenP(B) = 75/1000 = .075.


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The Complement Rule

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

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

The Complement Ruleis used to determine theprobability of an event occurring by subtracting the

probability of the eventnotoccurring from 1.


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The Complement Rule continued

A~A

A Venn Diagramillustrating the complement rulewould appear as:


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Example 4

If Dis 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 Cis the event that a

flight arrives on time, then

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


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Example 4 continued

C

.8

D

.025

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

.175

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

= 1 - [.8 +.025]

=.175


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The General Rule of Addition

If Aand Bare two

events that are notmutually exclusive,

then P(Aor B) is

given by the

following formula:

P(Aor B) = P(A) + P(B) - P(Aand B)


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The General Rule of

Addition

A and B

A

B

The Venn Diagram illustrates this rule:


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EXAMPLE 5

Stereo

320

Both

100

TV

175

In a sample of 500 students, 320 said they had a

stereo, 175 said they had a TV, and 100 said they hadboth. 5 said they had neither.

If t d t i l t d t


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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

r a n d o m , w h a t i s t h e

probability that the student

has only a stereo or TV?What is the probability

that the student has both a

s t e r e o a n d T V ?


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Joint Probability

A Joint Probabilitymeasures the likelihoodthat two or more events will happen concurrently.

An example would

be the event that astudent has both a

stereo and TV in his

or her dorm room.


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Special Rule of Multiplication

Two events Aand Bare independentif the

occurrence of one has no effect on the probability ofthe occurrence of the other.

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

The Special Rule of Multiplication

requires that two eventsA andBareindependent.


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Example 6

5-year stock prices

05

10

15

20

25

30

3540

45

1 2 3 4 5

Year

Stock

price

$

IBM

GE

P(I BMand GE) = (.5)(.7) = .35

Chris owns two stocks,

IBM and General

Electric (GE). Theprobability that IBM

stock will increase in

value next year is .5 and

the probability that GEstock will increase in

value next year is .7.

Assume the two stocks

are independent. Whatis 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)

What

is theprobability

that at least

one of these stocks

increases in value in

the next year?

This means that

either one can

increase or

both.

(.5)(1-.7)

+ (.7)(1-.5)

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


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Conditional Probability

The probability ofevent Aoccurring

given that the event

Bhas occurred iswritten P(A |B).

A Conditional Probabilityis theprobability of a particular event occurring,given that another event has occurred.


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General Multiplication Rule

It states that for two

eventsAandB, thejoint probability that

both events will happen

is found by multiplying

the probability thateventAwill happen by

the conditional

probability ofBgiven

thatAhas occurred.

The General

Rule ofMultiplicationisused to find the joint

probability that twoevents will occur.


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General Multiplication Rule

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

following formula:

P(Aand B) = P(A)P(B/A)

or

P(Aand 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 afemale, what is the

probability that she is an

accounting major?

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Tree Diagrams

Example 8: In a bag

containing 7 redchips and 5 blue

chips you select 2

chips one after the

other without

replacement.

Construct a tree

diagram showing thisinformation.

A Tree Diagramis

useful forportraying

conditional and

joint probabilities.

It is particularlyuseful for

analyzing business

decisions involving

several stages.

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Example 8 continued

R1

B1

R2

B2

R2

B2

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Bayes Theorem

)/()()/()(

)/()()|(

2211

11

1 ABPAPABPAP

ABPAPBAP

Bayes Theorem is a method

for revising a probabilitygiven additional information.

It is computed using the

following formula:

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Example 9

Duff Cola Company

recently received several

complaints that their bottlesare under-filled. A

complaint was received

today but the production

manager is unable to

identify which of the two

Springfield plants (AorB)

filled this bottle. What is

the probability that the

under-filled bottle came

from plantA?

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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

AUPAP

UAP

The likelihood the bottle was filled in

PlantAis 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 shirtand tie outfitsdoes he have?

The Multiplication

Formula indicatesthat if there are mwaysof doing one thing and n

ways of doing another

thing, there are mxn

ways of doing both.

(10)(8) = 80

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)!(

!

rn

nP

rn

A Permutationis any arrangement of robjects selected from npossible objects.

Note: The order of arrangement is important in

permutations.

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)!(!

!

rnr

nCrn

ACombinationis the number of

ways to choose r

objects from a

group of nobjectswithout regard to

order.

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Example 11

792

)!512(!5

!12512

C

There are 12 players

on the Carolina

Forest High Schoolbasketball team.

Coach Thompson

must pick five

players among thetwelve 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 thegroup, he must

also rank each of

the players in

that starting

lineup according

to their ability

(order matters).

Chapter 05 A Survey of Probability Concepts

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