Which factory made the defective part?

Question

Factory A makes 70\% of a company's parts with a 1\% defect rate. Factory B makes the other 30\% with a 4\% defect rate. A randomly chosen part is defective. What is the probability it came from Factory B?

Accepted Answer

Apply Bayes. The prior is P(A) = 0.70, P(B) = 0.30. The defect likelihoods are P(	ext{def} \mid A) = 0.01 and P(	ext{def} \mid B) = 0.04. Then P(B \mid 	ext{def}) = \frac{P(B)\,P(	ext{def} \mid B)}{P(A)\,P(	ext{def} \mid A) + P(B)\,P(	ext{def} \mid B)} = \frac{0.30 	imes 0.04}{0.70 	imes 0.01 + 0.30 	imes 0.04}. The numerator is 0.012 and the denominator is 0.007 + 0.012 = 0.019, so P(B \mid 	ext{def}) = \frac{0.012}{0.019} \approx 0.632 = 63.2\%. Even though Factory B makes only 30\% of parts, it produces about 63\% of the defective ones, because its defect rate is four times highe…

Which factory made the defective part?

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