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Squared streak payout

Expected valueMedium

A fair coin is flipped repeatedly and independently until the first tail. Let $K$ be the number of heads before that tail. You are paid $K^{2}$ . What is your expected payout?

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Hint

Condition on the first flip: after a head the process restarts with the count shifted by $1$ .

Solution

Let $m = E [K^{2}]$ and note $E [K] = 1$ for this geometric count.
With probability $\frac{1}{2}$ the first flip is a tail and $K = 0$ .
With probability $\frac{1}{2}$ it is a head, and then $K = 1 + K^{'}$ , where $K^{'}$ is a fresh copy of $K$ .
So $m = (\frac{1}{2}) \times E [(1 + K^{'})^{2}] = (\frac{1}{2}) \times (1 + 2 \times 1 + m)$ .
This gives $m = \frac{3}{2} + \frac{m}{2}$ , hence $m = 3$ .

Answer: $3$

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