Question

A manufacturing process produces integrated circuit chips. Over the long run, the fraction of bad chips produced by the process is 20%. Thoroughly testing a chip to determine whether it is good or bad is rather expensive, so a cheaper test is used. All good chips will past the test, but so will 7.5% of the bad chips. (a) Given that a chip passes the test, what is the probability that it is a good chip? Use at least 3 decimal places. (b) If the company sells all chips that pass the cheaper test, what percentage of sold chips will be bad? Use at least 3 decimal places.

Answer 1

(a) Given that a chip passes the test, what is the probability that it is a good chip?

LetB = {the chip is good}

A={the chip passes the cheap test}.

Bc={the chip is bad}

Ac={the chip fails the cheap test}

P(A | B) = 1

P(A | B c ) = 0.075

=
`(P(A | B)P(B))/(P(A | B)P(B) + P(A | Bc)P(Bc))` =
`(1.0.8)/(1.0.8+ 0.075 · 0.2)` ≈ 0.9751

(b) If the company sells all chips that pass the cheaper test, what percentage of sold chips will be bad?

P(B c |A) = 1 − P(B | A) = 1 - 0.9751 = 0.0249