Doubling coin prize, capped after four flips

Question

You flip a fair coin. If the first heads lands on flip number $n$ , you win $2^n$ dollars ( $2, then$ 4, then $8, doubling each flip). The game runs at most 4 flips. If all 4 flips are tails, you still collect the maximum prize of$ 16.

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: you need n-1 tails then a head, probability (1/2)^n. Each level contributes 1. The prize on level n is 2^n and its chance is (1/2)^n, so the product is: \left(	frac{1}{2}ight)^n 	imes 2^n = 1. The shrinking chance exactly cancels the growing prize, so every level is worth 1. Add the levels. Heads can first appear on flip 1, 2, 3, or 4, giving four contributions of 1 each, so 4. The cap. All 4 flips tails has probability (1/2)^4 = 1/16, and pays 16, adding: \frac{1}{16} 	imes 16 = 1. Total: EV =…

Doubling coin prize, capped after four flips

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