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PROBABILIT

Y:

Ngee Ann PolytechnicSchool of BA

CONCEP

TS

3- 2A Survey of Probability ConceptsOUTLINE

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

1. Define probability.2. Understand the terms: experiment, event and

outcome,3. Describe the classical, empirical, and

subjectiveapproaches to probability.

4. Calculate probabilities applying the rules ofaddition and the rules of multiplication.

5. Define the terms: conditional probability and joint probability.

6. Use a tree diagram to organize and compute probabilities.

7. Calculate a probability using Bayes’ theorem

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A Survey of Probability Concepts

OBJECTIVES

By the end of this chapter, you will be able to:12

Define the basic terms used in probability. Distinguish between mutually and non-mutually exclusive events; independent and dependent events. Compute single (marginal), joint and conditional probabilities

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Chance, Consequence & Strategy:Likelihood or Probability

Since there is little in life that occurs with absolute certainty, probability theory has found application in virtually every field of human endeavor.

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Definitions

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.

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Definitions continued

An experiment is the observation of some activityor the act of taking some measurement.

An outcome is the particular result of anexperiment.

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

experiment.

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Mutually Exclusive Events

Events are mutually exclusive if theoccurrence of any one event means that none of the otherscan occur at the same time.

Events are independent if the occurrence of one

event does not affect the occurrence of another. Events are collectively exhaustive if at least

one of the events must occur when an experiment is conducted.

Multiplicative rule

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Definitions

There are three definitions of probability: classical, empirical, and subjective.•The classical definition applies when there are n equallylikely 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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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 1

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Example 2

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EXAMPLE 3Throughout 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 anA:

P( A) 186 0.1551200

Expected Outcomes

Total possible Outcomes

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Basic 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) AB

P(A and B) = 0 Mutually exclusive

3- 13EXAMPLE 4 New England Commuter Airways recently supplied the following information

on their commuter flights from Boston to New York:

A r r i v a l F r e q u e n c y E a r l y

1 0 0

O n T i m e 8 0 0 L a t e

7 5

C a n c e l e d 2 5To t a l 1 0 0 0

EventAC B D

If A is the event that a flight arrives early, then

P(A) = 100/1000 = 0.10

If B is the event that a flight arrives late, then

P(B) = 75/1000 = 0.075

The probability that a flight is either earlyor

late is:

P(A or B) = P(A) + P(B) = .10 + .075 =0.175

3- 14Pg 107 Q14(a)The chair of the Board of Directors says, “ There is 50% chance this company will earn a profit, a 30% chance it will break even, and a 20% chance it will lose money next quarter”a. Use the addition rule to find probability the company will not lose money next quarter?

Profit= P , Even = E , Loss = L

P(P) = 0.5

P(E)=0.3

P(L)=0.2

Will not lose money = Make Profit or Even

or = ⋃P(P or E)

=P(P⋃E)=P(P)+P(E) = 0.5+0.3 =0.8

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The 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) isthe complement of A,

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

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The Complement Rule continued

A Venn diagram illustrating the complement rulewould appear as:

A

~A

P(A) + P(~A) = 1

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EXAMPLE 5

Recall EXAMPLE 4. Use the complement rule to findthe 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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EXAMPLE 5 continued

P(A or B) + P(C or D) =1000/1000= 1

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

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

C.8

D.025

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

3- 19Pg 107 Q14(b)

The chair of the Board of Directors says, “ There is 50% chance this company will earn a profit, a 30% chance it will break even, and a 20% chance it will lose money next quarter”

b. Use the complement rule to find probability thecompany will not lose money next quarter?

Profit= P ,Even = E ,Loss

= L P(P) = 0.5

P(E)=0.3

P(L)=0.2

Will not lose money = ~L

P(L)+P(~L)=1

P(~L)=1 - P(L) = 1-0.2

=0.8

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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:

Not Mutually exclusive

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

P(A U B) = P(A) + P(B) - P(A ⋂ B)

P(A U B) = P(A) + P(B)

Mutually exclusive

P(A ⋂ B)=0

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The General Rule of Addition

The Venn Diagram illustrates this rule:

A and B

B

A

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EXAMPLE 6

In a sample of 500 students, 320 said they had a stereo, 175 said they had a TV only, and 100 (including the 320 & 175 mentioned above) said they had both:

Stereo320

Both 100

TV 175

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EXAMPLE 6 continued

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

P(S) = 320/500 = 0.64.

P(T) = 175/500 =0.35.

P(S and T) = 100/500 = 0.20If 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 =0.79

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Pg108 Q22 A study by the National Park Service revealed that 50% of

vacationers going to the Rocky Mountain region visit Yellowstone Park, 40% visit Tetons, and 35 % visit both.

a) What is the probability that a vacationer will visit at least one of these attractions?

Yellowstone= Y

Tetons =TP(Y)=0.5, P(T)= 0.4Both =and=⋂P(Y ⋂ T)=0.35At Least = or = ⋃P(Y ⋃ T)=???P(Y ⋃ T) = P(Y) + P(T) - P(Y and T)

= 0.5+0.4-0.35

=0.55

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Special 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)

3- 26Page 130 Q62Forty percent of the homes constructed in the Quail Creek area include a security system. Three homes are selected at random.

a) What is the probability all three homes have a security system?b) What is the probability none of the three selected homes have a security system?c) What is the probability at least one of the selected homes has a security system?

S = Home has a security systemP(S) = 0.4 P(~ S) = 0.6a) P(SSS) = P(S)*P(S)*P(S)

= 0.4*0.4*0.4 = 0.064b) P(~S~S~S) = P(~S) *P(~S)*P(~S)

= 0.6*0.6*0.6 = 0.216c) P(at least one) = 1- P(none has a security system)

=1 -0.216 =0.784

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Example : Pg 117 Q 28Three defective electric toothbrushes were accidentally shipped to a drugstore by Cleanbrush Products along with 17 nondefective ones.

a. What is the probability the first two electric toothbrushes

sold will be returned to the drugstore because they are

defective?

b.What is the probability the first two electric toothbrushes sold

will not be defective?D=defective ~D= Not defective

N=20

a. P(DD)= 3/20*2/19 = 6/380

b. P(~D~D)= 17/20 * 16/19=272/380

3- 28EXAMPLE 7Chris owns two stocks, IBM and General Electric (GE). The probability that IBM stock will increase in value next year is .5 and theprobability 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 ) = 0.5 P(~IBM ) =0.5P(GE) = 0.7 P(~ GE) = 0.3

P(IBM and GE) = (.5)(.7) = 0.35 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) = P(IBM)P(~GE)+P(~IBM)P(GE)+P(IBM)P(GE) P(at

least one) = (.5)(.3) + (.5)(.7) + (.7)(.5) = 0.85

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Joint Probability

A joint probability measures the likelihood that two ormore 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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Conditional 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 theevent B has occurred is written P(A|B).

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General 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 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 8

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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EXAMPLE 8 continued

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 Diagrams

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

EXAMPLE 9: 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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EXAMPLE 9 continued

R1

B1

R2

B2

R2

B2

7/12

5/12

6/11

5/11

7/11

4/11

7 Red 5 Blue Total = 12

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Bayes’ Theorem

Bayes’ Theorem is a method for revising a probabilitygiven additional information.

It is computed using the following formula:

P(A1 )P(B / A1 ) P(A2 )P(B / A2 )

P(A1 )P(B / A1 )

P(A | B)

1

A1A2

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Bayes’ Theorem

P( A1 )P(B / A1 ) P( A2 )P(B / A2 ) P( A3 )P(B / A3 )

P( A1 )P(B / A1 )

P( A | B)

1

P( A1 )P(B / A1 ) P( A2 )P(B / A2 )...... P( AK )P(B / AK )

P( A1 )P(B / A1 )

P( A | B) 1

3 Events

K Events

3- 39EXAMPLE 10Duff 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?

The following table summarizes the Duff production experience.

% of Total Production

% of under- filled bottles

A 55 3

B 45 4

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Example 10 continued

P ( A / U ) P ( A ) P (U / A )

P ( A ) P (U / A ) P ( B ) P (U / B )

.5 5 (. 0 3)

. 4 7 8 3

.5 5 (.0 3) .4 5 (.0 4 )

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

P(A) =0.55

P(B) =0.45P(U/A)=0.03

P(U/B)=0.04

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A

BD

U2

U

~U

0.55

0.45

0.03

004

0.55*0.03 = 0.0165

0.45* 0.04=0.018

Total = 0.0345

P(A/U)= 0.0165/0.0345=0.4783

~U

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3- 42Example : Pg 121 Q 35The Ludlow Wildcats baseball team, a minor league team in the

Cleveland Indians organization, plays 70 percent of their games at night and 30 percent during the day. The team wins 50 percent of their night games and 90 percent of their day games. According to today's newspaper, they won yesterday. What is the probability the game was played at night?

N = Night

D= Day

W= Win

P(N) = 0.7

P(D) = 0.3

P( W / N) = 0.5

P( W / D) = 0.9

P( N / W) =?????

Find Night / Given: won yesterday

Win / Given

night Win / Given

Day

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Example : Pg 171 Q 35P( N) = 0.7 P( D) = 0.3

P( W / N) = 0.5 P( W / D) = 0.9

= 0.7 *0.50.7 *05 + 0.3

*0.9

P( N / W) =

P(W/N) * P(N)P(W/N) * P(N) + P(W/D) *

P(D)

=0.5645

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Tree Diagram

N

DD

W2

W

~W

0.7

0.3

0.5

0.9

0.7*0.5 = 0.35

0.3 * 0.9=0.27

Total = 0.62

P(N/W)= 0.35/0.62=0.5645

~W

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SummaryoIntersection - and, both

or,at least

oUnion -oMutually exclusive: AdditionruleoNon Mutually exclusive:o Addition ruleoIndependent –Multiplicative ruleoDependent –Baye’s Theorem

P(A B) P( A) P(B)

P(A B) P(A) P(B) P(A B)

P( A1 | B) P(A1)P(B | A1) P(A1)P(B | A1) P(A2)P(B |

A2) ...P(An)P(B | An)

P(ABC ) P(A)* P(B)* P(C)