Stat Chapter 4 c

8/13/2019 Stat Chapter 4 c

1/17

Bayes Theorem

Suppose we have estimatedprior probabilities for eventswe are concerned with, andthen obtain new information.

We would like to a soundmethod to computed revised

or posterior probabilities.Bayes theorem gives us a

way to do this.


2/17

Probability Revision using Bayes

Theorem

PriorProbabilities

NewInformation

Application of

Bayes Theorem

PosteriorProbabilities


3/17

Application of Bayes Theorem

Consider a manufacturing firm that receivesshipment of parts from two suppliers.

Let A1 denote the event that a part is receivedfrom supplier 1; A2 is the event the part isreceived from supplier 2


4/17

We get 65 percent of our parts

from supplier 1 and 35percent from supplier 2.

Thus:

P(A 1) = .65 and P(A 2) = .35


5/17

Quality levels differ between suppliersPercentageGood Parts

PercentageBad Parts

Supplier 1 98 2

Supplier 2 95 5

Let G denote that a part is good and B denotethe event that a part is bad. Thus we have thefollowing conditional probabilities:

P (G | A1 ) = .98 and P (B | A2 ) = .02

P (G | A2 ) = .95 and P (B | A2 ) = .05


6/17

Tree Diagram for Two-Supplier Example

Step 1Supplier

Step 2Condition

ExperimentalOutcome

A1

A2

G

B

G

B

(A1, G)

(A1, B)

(A2 , G)

(A2 , B)


7/17

Each of the experimentaloutcomes is the intersection of 2

events. For example, theprobability of selecting a partfrom supplier 1 that is good is

given by:

)|()()(),( 1111 AG P A P G A P G A P


8/17

Probability Tree for Two-Supplier Example

6370.)|()()( 111 AG P A P G A P

0130.)|()()( 111 A B P A P B A P

3325.)|()()( 222 AG P A P G A P

Step 1Supplier

Step 2Condition

Probability of Outcome

P(A 1)

P(A 2 )

.65

.35

P(G | A 1 )

P(B | A2 )

P(B | A 2 )

P(B | A 2 )

.98

.95

.02

.050175.)|()()( 222 AG P A P B A P


9/17

A bad part broke oneof our machines sowere through for the

day. What is theprobability the partcame from suppler 1?

We know from the law of conditional probability that:

)()(

)|( 11 B P B A P

B A P

Observe from the probability tree that:

)|()()( 111 A B P A P B A P

(4.14)

(4.15)


10/17

The probability of selecting a badpart is found by adding togetherthe probability of selecting a bad

part from supplier 1 and theprobability of selecting bad part

from supplier 2.

That is:

)/()()|()(

)()()(

2211

21

A B P A P A B P A P

B A P B A P B P (4.16)


11/17

Bayes Theorem for 2 events

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

2211

111 A B P A P A B P A P

A B P A P B A P

By substituting equations (4.15) and (4.16) into(4.14), and writing a similar result for P(B | A 2 ), weobtain Bayes theorem for the 2 event case:

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

)|(2211


A B P A P

B A P


12/17

Do the Math

4262.0305.0130.

)05)(.35(.)02)(.65(.)02)(.65(.

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

2211


A B P A P B A P

5738.0305.0175.

)05)(.35(.)02)(.65(.)05)(.35(.

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

)|(2211


A B P A P B A P


13/17

Bayes Theorem

)|()(...)|()()|()()|()(

)|(2211 nn

iii

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

A B P A P B A P


14/17

Tabular Approach to Bayes Theorem 2-Supplier Problem

(1)

EventsA i

(2)Prior

ProbabilitiesP(A i )

(3)ConditionalProbabilities

P(B | A 1 )

(4)Joint

ProbabilitiesP(A i B)

(5)Posterior

ProbabilitiesP(A i | B)

A1 .65 .02 .0130 .0130/.0305=.4262

A2 .35 .05 .0175 .0175/.0305

=.57381.00 P(B)=.0305 1.0000


15/17

Using Excel to ComputePosterior Probabilities

Prior Conditional Joint Posterior

Events Probabilities Probabilities Probabilities Probabilities

A 1 0.65 0.02 0.013 0.426229508

A 2 0.35 0.05 0.0175 0.573770492

0.0305 1.0000


16/17

Exercise 41, p. 187

41. A consulting firm submitted a bid for a large consultingcontract. The firms management felt id had a 50 -50 changeof landing the project. However, the agency to which the bidwas submitted subsequently asked for additionalinformation. Past experience indicates that that for 75% ofsuccessful bids and 40% of unsuccessful bids the agency

asked for additional information.a. What is the prior probability of the bid being successful

(that is, prior to the request for additional information).

b. What is the conditional probability of a request for

additional information given that the bid will beultimately successful.

c. Compute the posterior probability that the bid will besuccessful given a request for additional information.


17/17

Exercise 41, p. 187Let S 1 denote the event of successfully obtaining the project.

S 2 is the event of not obtaining the project.B is the event of being asked for additional information about a bid.

a. P(S 1 ) = .5

b. P(B | S 1 ) = .75

c. Use Bayes theorem to compute the posterior probability that arequest for information indicates a successful bid.

652.575.375.

)4)(.5(.)75)(.5(.)75)(.5(.

)()()(

)|(21

11 BS P BS P

BS P BS P

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