Each archer's accuracy is random, then sixteen shots

Question

A randomly chosen archer has a hit probability $p$ that is spread evenly between $0.5$ and $1$ (uniform on $[0.5, 1]$ ). That archer then takes $16$ independent shots at the target.

How many hits should you expect?

Accepted Answer

First the archer (and thus the hit probability p) is drawn, then 16 shots follow. The tower rule separates the layers: E[	ext{hits}] = E\big[\,E[	ext{hits} \mid p]\,\big]. Step 1, freeze the archer. Given hit probability p, sixteen shots average E[	ext{hits} \mid p] = 16\,p. Step 2, average over the archer. A uniform draw on [0.5, 1] has mean equal to the midpoint, E[p] = \frac{0.5 + 1}{2} = 0.75. So E[	ext{hits}] = E[16\,p] = 16\,E[p] = 16 	imes 0.75 = 12. You expect 12 hits: sixteen shots at an average accuracy of three-quarters. Takeaway. When the shooter's accuracy is itself random, …

Each archer's accuracy is random, then sixteen shots

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