A company that produces a particular machine component has 3

A company that produces a particular machine component has 3 factories, one each in Buffalo, Dayton, and Pittsburgh. 29% of the components produced come from the Buffalo factory, 32% of the components come from the Dayton factory, and 39% of the components come from the Pittsburgh factory. It is known that 1.9% of the components from the Buffalo factory, 1.4% of the components from the Dayton factory, and 1.2% of the components from the Pittsburgh factory are defective. Given that a component is selected at random and is found not to be defective, what is the probability that the component was made in Dayton?

Solution

Let X = the event that the component is defective
D, B, P = the events that they came from Dayton, Buffalo, and Pittsburgh, respectively.

Note that

P(D|not X) = P(not X and D) / P(not X)

By Bayes\' Rule,

P(X) = P(X|B) P(B) + P(X|D) P(D) + P(X|P) P(P)

Thus,

P(X) = 0.01467

Thus,

P(not X) = 0.98533

Also,

P(not X and D) = P(not X|D) P(D) = [1 - P(X|D)] P(D) = 0.31552

Thus,

P(D|not X) = P(not X and D) / P(not X)

= 0.31552 / 0.98533

= 0.32022 [ANSWER]