A company has 100 factories all of which produce the same nu

A company has 100 factories, all of which produce the same number of products. In 99 of these factories, only 0.01% of the products are defective. In 1 of the factories, however, the rate of defect is 2%. Suppose that you have just purchased one of this companys products and nd that it is defective. What is the probability that it comes from the bad factory?

Solution

Let

B = bad factory
D = defective

Thus, as

P(B|D) = P(B) P(D|B) / P(D)

As

P(D) = P(B) P(D|B) + P(B\') P(D|B\') = (1/100)*(0.02) + (1-1/100)*(0.0001) = 0.000299

Thus,

P(B|D) = P(B) P(D|B) / P(D) = (1/100)*0.02 / 0.000299

= 0.668896321 [ANSWER]