Draw a count, then flip that many coins

Question

A jar holds tickets numbered 1, 2, 3, 4, 5. You draw one at random and call it N. You then flip N fair coins and count how many land heads. What is the expected number of heads?

Accepted Answer

Two layers of randomness sit here: first the ticket N, then the coin flips. Handle them with the tower rule: E[	ext{heads}] = E\big[\,E[	ext{heads} \mid N]\,\big]. Step 1, freeze the ticket. If you know N = n, you flip n fair coins. Each head shows up with probability 	frac12, so on average E[	ext{heads} \mid N = n] = 	frac12\,n. Step 2, average over the ticket. The ticket is uniform on \{1,2,3,4,5\}, so E[N] = \frac{1+2+3+4+5}{5} = 3. Plugging the Step 1 answer into the tower rule, E[	ext{heads}] = E\big[	frac12 N\big] = 	frac12\,E[N] = 	frac12 	imes 3 = 1.5. You expect 1.5 heads. …

Draw a count, then flip that many coins

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