Four percent of items produced by a production process are defective, and the re

Question

Four percent of items produced by a production process are defective, and the remaining 96% of items produced are not defective. The items are subjected to a manual inspection procedure. The inspection procedure is not foolproof: in fact 10% of defective items are accepted by the manual inspection procedure while 5% of the non-defective items are rejected by it.

(i) Find the probability that a randomly selected item is accepted by the inspection procedure.

(ii) An item has been rejected by the inspection procedure - what is the probability that it is defective?

Explanation / Answer

i. P( item is accepted ) = P(item is defective) * P(item is accepted | item is defective) + P(item is not defective) * P(item is accepted | item is not defective) = 0.04*0.10 + 0.96*(1-0.05) = 0.9160

ii. P(item is rejected) = P(item is defective) * P(item is rejected | item is defective) + P(item is not defective) * P(item is rejected | item is not defective) = 0.04*(1-0.10) + 0.96*0.05 = 0.0840

By Bayes' theorem,

P(item is defective | item is rejected) = P(item is defective) * P(item is rejected | item is defective) / P(item is rejected)

= 0.04*(1-0.10) / 0.0840 = 0.4286

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