Which factory made the defective part?

Question

A company sources one component from two plants. Plant A makes 70% of the components and 2% of them are defective. Plant B makes the other 30%, but 5% of its output is defective. A warehouse worker pulls a component at random and finds it defective.

What is the probability the defective component came from Plant B?

Accepted Answer

Let B be "from Plant B" and D be "defective." We want P(B \mid D). Picture 1000 components. - Plant A: 700 made, 2\% defective \Rightarrow 14 defective. - Plant B: 300 made, 5\% defective \Rightarrow 15 defective. Total defectives: 14 + 15 = 29, of which 15 came from B: P(B \mid D) = \frac{15}{29} \approx 0.517. So a defective part is slightly more likely to be from the smaller plant, because its higher defect rate outweighs its smaller share of production. Takeaway. Bayes weighs each source by volume times defect rate, not by volume alone. A minority supplier with a bad quality record can do…

Which factory made the defective part?

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