A component is produced on one of 4 production lines. All production lines will

Question

A component is produced on one of 4 production lines.  All production lines will produce the parts at the same rate, i.e. probability of a single part coming from any one production line is 25%.  Historically, each production line has a history of producing parts at a 2.5% defect rate.

Production Line 1 produced parts for a short period of time with a 5% defect rate.  Unfortunately, the manufacturer shipped the parts before learning of the issue on Production Line 1.  The customer tested 3 of the fuses and found 1 defect.

What is the probability that the lot of parts came from Production Line 1?

(Hint: You will also need to use the Binomial distribution to determine the conditional probability of the defective parts coming off the different production lines, e.g.:

PDLine 1=bx;n,p=0.05=nxpxqn-x)

PDLines 2,3,4=bx;n,p=0.025=nxpxqn-x

Explanation / Answer

Binomial Distribution

nCx pxq(n-x)

Line 1: p=0.05

Line 2,3,4: p=0.025

Probability of finding 1 defect in 3 fuses in Line 1

3C1(0.05)1*(0.95)2 = 0.135

Probability of finding 1 defect in 3 fuses in Line 2,3,4

3C1(0.025)1*(0.975)2 = 0.071

Every part has a 25% probability of coming from any of the production lines 1,2,3,4.

Total probability of finding 1 defect in 3 fuses

= (0.25*0.135) + (0.25*0.071) + (0.25*0.071) + (0.25*0.071)

= 0.087

Conditional Probability

P(A|B) = P(A B) / P(B)

A - Event that the lot of parts came from Production Line 1

B - Event that 1 out of 3 fuses are defective

P(A|B) = P(finding 1 defect in 3 fuses and parts are from Line 1) / P(finding 1 defect in 3 fuses)

= (0.135*0.25)/0.087

= 0.388

The probability that the lot of parts came from Production Line 1 is 0.388.

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