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.

Given that a chip passes the test, what is the probability that the chip was defective?

0.0244 (2.44%)

Step-by-step explanation:

defining the event T = the chips passes the tests, then

P (T) = probability that the chip is not defective * probability that it passes the test given that is not defective + probability that the chip is defective * probability that it passes the test given that is defective = 0.80 * 1 + 0.20 * 0.10 = 0.82

for conditional probability we can use the theorem of Bayes. If we define the event D=the chip was defective, then

P (D/T) = P (D∩T) / P (T) = 0.20 * 0.10/0.82 = 0.0244 (2.44%)

where

P (D∩T) = probability that the chip is defective and passes the test

P (D/T) = probability that the chip is defective given that it passes the test