A coin of random bias, then twenty flips

Question

A coin is manufactured so that its probability of heads, $p$ , is a random number spread evenly between $0$ and $1$ (uniform on $[0,1]$ ). Once $p$ is fixed, you flip that coin $20$ times.

How many heads should you expect? As a follow-up, what if $p$ is instead spread evenly between $0.2$ and $0.6$ ?

Accepted Answer

There are two layers: first the hidden bias p is drawn, then 20 flips happen with that bias. The tower rule handles it: E[	ext{heads}] = E\big[\,E[	ext{heads} \mid p]\,\big]. Step 1, freeze the bias. If you knew the coin's bias were exactly p, then 20 flips give a Binomial count with mean E[	ext{heads} \mid p] = 20\,p. Step 2, average over the bias. Now let p be random again. For a uniform draw on [0,1] the average is E[p] = 0.5. So E[	ext{heads}] = E[20\,p] = 20\,E[p] = 20 	imes 0.5 = 10. You expect 10 heads, exactly as if the coin were simply fair. The randomness in p does not change t…

A coin of random bias, then twenty flips

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