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. 24% of the components produced come from the Detroit factory, 36% of the components come from the Pittsburgh factory, and 40% of the components come from the Cleveland factory. It is known that 2.0% of the components from the Detroit factory, 1.5% of the components from the Pittsburgh factory, and 2.6% of the components from the Cleveland factory are defective. Given that a component is selected at random and is found to be defective, what is the probability that the component was made in Cleveland?

Explanation / Answer

P(Detroit) = 0.24

P(Pittsburgh) = 0.36

P(Cleveland) = 0.4

P(defective | Detroit) = 0.02

P(defective | Pittsburgh) = 0.015

P(defective | Cleveland) = 0.026

P(defective) = P(defective | Detroit) * P(Detroit) + P(defective | Pittsburgh) * P(Pittsburgh) + P(defective | Cleveland) * P(Cleveland)

                    = 0.02 * 0.24 + 0.015 * 0.36 + 0.026 * 0.4

                    = 0.0206

P(Cleveland | Defective) = P(defective | Cleveland) * P(Cleveland) / P(defective)

                                       = 0.026 * 0.4 / 0.0206

                                       = 0.5049

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