Doubling coin prize, but three flips or nothing

Question

You flip a fair coin. If the first heads lands on flip number $n$ , you win $2^n$ dollars ( $2, then$ 4, then $8). But the game runs only 3 flips. If none of the 3 flips is heads, you win nothing at all.

What is the fair value of this game?

Accepted Answer

Add up each prize times its chance, using the doubling trick. Chance of first heads on flip n: n-1 tails then a head, probability (1/2)^n. Each level contributes 1. Prize 2^n times chance (1/2)^n equals: \left(	frac{1}{2}ight)^n 	imes 2^n = 1. Add the levels. Heads can first appear on flip 1, 2, or 3, giving three contributions of 1 each, so 3. The "nothing" case. All 3 flips tails has probability (1/2)^3 = 1/8, but here it pays 0, so it adds nothing: \frac{1}{8} 	imes 0 = 0. Total: EV = 3 + 0 = 3. This is the difference between a cap and a cliff. Here, failing to get heads pays zero rat…

Doubling coin prize, but three flips or nothing

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