Random defect rate, then inspect twelve items

Question

A machine's defect rate for the day, p, is a random number spread evenly between 0 and 1 (you have no idea whether the machine is running well or poorly). Once the rate is set, you inspect 12 items produced that day, each defective independently with probability p. How many defective items should you expect?

Accepted Answer

Two layers: the rate p is drawn first, then 12 items come out at that rate. The tower rule separates them: E[	ext{defects}] = E\big[\,E[	ext{defects} \mid p]\,\big]. Step 1, freeze the rate. If today's rate were exactly p, then 12 items average E[	ext{defects} \mid p] = 12\,p. Step 2, average over the rate. A uniform draw on [0,1] has mean E[p] = 0.5, so E[	ext{defects}] = E[12\,p] = 12\,E[p] = 12 	imes 0.5 = 6. You expect 6 defective items. Complete ignorance about the rate (uniform on [0,1]) behaves, on average, like a known rate of one-half. Takeaway. A completely unknown rate spread …

Random defect rate, then inspect twelve items

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