A fair coin is flipped until it shows heads, up to a maximum of 4 flips. You are paid $10 for every flip that occurs. What is the fair price of the game?

A fair coin is flipped until it shows heads, up to a maximum of 4 flips. You are paid $10 for every flip that occurs. What is the fair price of the game?

Approach: Write down the probability of each stopping point, noting where the cap collects the remaining mass, then use the tail sum form of the expectation as a cross-check.

$18.75. Let N be the number of flips. The run stops at the first head and is capped at 4, so P(N = 1) = 1/2, P(N = 2) = 1/4, P(N = 3) = 1/8, and N of 4 collects the remaining 1/8, covering both a head on the fourth flip and no head at all. The expected value of N is 1/2 + 2/4 + 3/8 + 4/8, which is 15/8 or 1.875, so the fair price is 10 * 1.875 = $18.75. The tail sum identity is faster, since E[N] is the sum over k of P(N >= k) = 1 + 1/2 + 1/4 + 1/8 = 15/8. No outcome weighting is needed at all. Uncapped, N follows a geometric distribution with mean 2, so the cap costs 0.125 flips, or $1.25.

Follow-up: The payoff becomes $10 times 2 to the power N minus one, with the same cap of 4 flips. What is the fair price, and what happens as the cap is lifted?

Key concepts: expected value, tail sum identity, geometric distribution.