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

Probability that a product is from Detroit factory = 0.28

Probability that a product is not defective and is from Detroit factory = (1-0.023)*0.28 = 0.27356

Probability that a product is from Pittsburgh factory = 0.41

Probability that a product is not defective and is from Pittsburgh factory = (1-0.019)*0.41 = 0.40221

Probability that a product is from Cleveland factory = 0.31

Probability that a product is not defective and is from Cleveland factory = (1-0.022)*0.31 = 0.30318

=> Probability that a non-defective component was made in Pittsburgh = 0.40221 / (0.27356+0.40221+0.30318)

= 0.41086

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