A manufacturing process produces integrated circuit chips. Over the long run the fraction of bad chips produced by the process is around 20%. Thoroughly testing a chip to determine whether it is good or bad is rather expensive, so a cheap test is tried. All good chips will pass the cheap test, but so will 10% of the bad chips. a. Given that a chip passes the test, what is the probability that the chip was defective? 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.

a. The probability that a chip is defective given that is passes the test is determined by the probability that a defective chip passes the test divided by the probability that a chip passes the test.

20% of the chips are bad, 10% of those chips pass the test:

[tex]P(Bad\ \&\ Pass) =0.20*0.10=0.02[/tex]

The probability that any chip passes is:

[tex]P(Pass) =P(Bad\ \&\ Pass) +P(Good\ \&\ Pass) \\P(Pass) =0.20*0.10+0.80*1.0\\P(Pass) =0.82[/tex]

The probability that a chip is defective given that is passes the test is:

[tex]P(Bad|Pass)=\frac{P(Bad\ \&\ Pass) }{P(Pass)}\\P(Bad|Pass)=\frac{0.02}{0.82}=0.0244[/tex]

The probability is 0.0244.

b. This question is asking for the same thing in as the previous item, just with different words, "If the company sells all chips that pass the cheaper test (Given that a chip passes the test), what percentage of sold chips will be bad ( what is the probability that the chip was defective)."

Therefore, the probability is also 0.0244