lecture on conditional & bayes_ rule.docx

7/27/2019 Lecture on conditional & Bayes_ rule.docx

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Conditional Probability,Bayes Theorem, and

Independence

Conditional Probability ExampleHousehold Survey on the plan of buying big-screen TV set.Sample Space = All Household SurveyedEvent A: Actually purchasedEvent B: Subject plans to buy big-screen TV

Interpret:Probability that a household has the plan to buy and actuallypurchased.Probability that a household has actually purchased althoughthere was no plan to buy.Probability that a household has not actually purchased

although that there was a plan to buy big-screen TV set.Probability that a household has actually purchased given thatthere was a plan to buy big-screen TV set.Probability that a household has actually purchased given thatthere was no plan to buy big-screen TV set.

( ) ( )

( )

( )

( )

( )'BAP

BAP

B'AP

'BAP

ABPBAP

=


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Law of Total Probability & Bayes RuleTake events A i for i = 1 to k to be:

Mutually exclusive: for all i,jExhaustive:

For any event B on S

Bayes theorem follows

SAA0AA

n1

ji=

=

)A(P)AB(P)A(P)AB(P)B(P nn11 ++=

)AP()AB(P...)AP()AB(P)AB(P)A(P

)AP()AB(P

)B(P

)BA(P)BA(P

nn2211

j j j

j +++=

=

A1 A2 A3

A4

An

B

ExampleOnly 1 in 400 bulbs produced by Factory A isdefective and 3 out of 600 bulbs produced byFactory B are defective

Factory A ND= 99.75%=0.9975 D=0.25%=0.0025Factory B, ND=99.5%=0.995 D=.5%=0.005If one randomly tested bulb is defective, what is theprobability that it is produced by Factory B?

Examine probabilitiesP( A) = 0.40P(B)= 0.60P(ND |A) = .9975, P(D| A)= 0.0025P(ND |B) = .995, P(D|B)= 0.005


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ExampleOnly 1 in 400 bulbs produced by Factory A isdefective and 3 out of 600 bulbs produced byFactory B are defective

Factory A ND= 99.75%=0.9975 D=0.25%=0.0025Factory B, ND=99.5%=0.995 D=.5%=0.005If one randomly tested bulb is defective, what is theprobability that it is produced by Factory B?

Examine probabilitiesP( A) = 0.40P(B)= 0.60P(ND |A) = .9975, P(D| A)= 0.0025

P(ND |B) = .995, P(D|B)= 0.005


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Assignment (Math 104)

1. An electronic fuse is produced by five production lines in a manufacturingoperation. The fuses are costly, are quite reliable, and are shipped tosuppliers in 100-unit lots. Because testing is destructive, most buyers of thefuses test only a small number of fuses before deciding to accept or rejectthe lots of incoming fuses.All the five production lines normally produce only 2 percent defective fuses,which are random dispersed in the output. Unfortunately, production line 1suffered mechanical difficulty and produced 5 percent defectives during themonth of March. This situation became known to the manufacturer after thefuses had been shipped. A costumer received a lot produced in March andtested three fuses. One failed. A) What is the probability that the lot wasproduced on line 1? B)What is the probability that the lot came from one of the four other lines?

2. The following table reflects the ending inventory of stocks of a certaindistributor company.

Name of supplier

(Company)

# of boxes of Pineapple juice

# of boxes of Orange juice

# of boxesGrape juice

A 100 320 300B 230 250 400

C 500 630 350

Based on the information given above, suppose that one box getsdestroyed due to mishandling, find the probability that;

a. Such box contained Orange juice.b. Such box contained Pineapple juice or supplied by company B.c. The box is supplied by Company C and contains orange juice.d. The box contained Pineapple juice given that it came from

Company A.e. Suppose that the box contained grape juice, what is the

probability that it was supplied by company C?


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3. An oil company purchased an option on land in Palawan. Preliminary geologicstudies assigned the following prior probabilities.

P(high-quality oil) =0.50

P(medium-quality oil) =0.20

P( no high-quality oil or no medium-quality oil) = 0.75

P(no oil) =0.30

Required:

a) What is the probability of finding oil?b) What is the probability of finding no high-quality oil?c) What is the probability of finding no medium-quality oil?d) What is the probability of finding no oil and medium-quality oil?

4. Of the gadgets produced by company A 30% are black color, while 20% of thenumber of gadgets manufactured by company B are non-black and 15% of the number of gadgets produced by company C are colored black. If onegadget is selected at random,

a. What is the probability that it is black given that the gadget isproduced by company B?

b. What is that probability that the gadget was produced by company B

given that it is black color?

lecture on conditional & bayes_ rule.docx

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