A defective bulb, which factory made it?

Question

A retailer buys light bulbs from three factories. Factory A supplies $50\%$ of the stock and $2\%$ of its bulbs are defective. Factory B supplies $30\%$ with a $3\%$ defect rate. Factory C supplies $20\%$ with a $5\%$ defect rate. A customer picks a bulb at random and it is defective.

What is the probability the defective bulb came from Factory A?

Accepted Answer

Imagine 1{,}000 bulbs. Split them by factory and count the defective ones each contributes. - Factory A: 500 bulbs, 2\% defective = 10. - Factory B: 300 bulbs, 3\% defective = 9. - Factory C: 200 bulbs, 5\% defective = 10. Total defective bulbs: 10 + 9 + 10 = 29. Of these, 10 came from Factory A: P(	ext{A} \mid 	ext{defective}) = \frac{10}{29} \approx 0.345. Even though Factory A has the lowest defect rate, it makes half of all bulbs, so it still accounts for a large share of the defects. Takeaway. To trace an effect back to a source, weight each source by both its volume (how much it suppl…

A defective bulb, which factory made it?

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