Doubling coin prize, capped after six flips

Question

You flip a fair coin. If the first heads lands on flip number $n$ , you win $2^n$ dollars ( $2,$ 4, $8, doubling each flip). The game runs at most 6 flips. If all 6 flips are tails, you still collect the maximum prize of$ 64.

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. The halving chance cancels the doubling prize, so every level is worth exactly 1. Add the levels. Heads can first appear on flip 1 through 6, giving six contributions of 1 each, so 6. The cap. All 6 flips tails has probability (1/2)^6 = 1/64 and pays 64, adding: \frac{1}{64} 	imes 64 = 1. Total: EV = 6 + 1 = 7. Each extra flip you allow adds exactl…

Doubling coin prize, capped after six flips

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