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.: PCDlLine 1) = b(x; n, p = 0.05) = C ) pan-x) P(D|Lines 2,3,4) = b (x; n, p = 0.025)-CD p"qn-r

Explanation / Answer

probability of a product from production line 1 =0.25

and probability of a product from other production lines=0.75

therefore probability that 1 out of 3 fuse is defective =P( product from production line 1 and 1 out of 3 fuse is defective+product from other production line and 1 out of 3 fuse is defective)

= 0.25*(3C1(0.05)1(0.95)2)+0.75*(3C1(0.025)1(0.975)2) =0.25*0.4219+0.75*0.1406 =0.2109

hence probability that lot came from production line 1 given 1 out of 3 is defective

=P(( product from production line 1 and 1 out of 3 fuse is defective)/P( 1out of 3 is defective)

==0.25*0.4219/(0.2109)=0.5

revert for any clarification required.

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