Doubling coin prize, capped after five flips

Question

You flip a fair coin. If the first heads lands on flip number $n$ , you win $2^n$ dollars: $2 for heads on flip 1,$ 4 on flip 2, $8 on flip 3, and so on, doubling each flip. But the game runs at most 5 flips. If all 5 flips are tails, you still collect the maximum prize of$ 32.

What is the fair value of this game?

Accepted Answer

Expected value means adding up each prize times its chance. The clever part is that this doubling game has a hidden pattern that makes the sum easy. The chance of first heads on flip n. To get your first heads exactly on flip n, you need n-1 tails followed by a head, which has probability (1/2)^n. So flip 1 has chance 1/2, flip 2 has chance 1/4, flip 3 has chance 1/8, and so on. Each term contributes the same amount. Look at the prize times its chance for a first heads on flip n: \left(	frac{1}{2}ight)^n 	imes 2^n = \frac{2^n}{2^n} = 1. Every level contributes exactly 1. This is the famou…

Doubling coin prize, capped after five flips

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