A company that produces a particular machine component has 3 factories, one each

Question

A company that produces a particular machine component has 3 factories, one each in Detroit, Pittsburgh, and Cleveland. 28% of the components produced come from the Detroit factory, 41% of the components come from the Pittsburgh factory, and 31% of the components come from the Cleveland factory. It is known that 2.3% of the components from the Detroit factory, 1.9% of the components from the Pittsburgh factory, and 2.2% of the components from the Cleveland 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 Pittsburgh?

Explanation / Answer

P(Detroit) = 0.28

P(Pittsburgh) = 0.41

P(Cleveland) = 0.31

P(defective | Detroit) = 0.023

P(not defective | Detroit) = 1 - 0.023 = 0.977

P(defective | Pittsburgh) = 0.019

P(not defective | Pittsburgh) = 1 - 0.019 = 0.981

P(defective | Cleveland) = 0.022

P(not defective | Cleveland) = 1 - 0.022 = 0.978

P(not defective) = P(not defective | Detroit) * P(Detroit) + P(not defective | Pittsburgh) * P(Pittsburgh) + P(not defective | Cleveland) * P(Cleveland)

                          = 0.977 * 0.28 + 0.981 * 0.41 + 0.978 * 0.31

                          = 0.979

P(Pittsburgh | not defective) = P(not defective | Pittsburgh) * P(Pittsburgh) / P(not defective)

                                              = 0.981 * 0.41 / 0.979

                                              = 0.4108

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