Tripling coin prize, capped after three flips

Question

You flip a fair coin. If the first heads lands on flip number $n$ , you win $3^n$ dollars: $3 for heads on flip 1,$ 9 on flip 2, $27 on flip 3. The game runs at most 3 flips. If all 3 flips are tails, you still collect the maximum prize of$ 27.

What is the fair value of this game?

Accepted Answer

Add up each prize times its chance. The twist here is that the prize triples while the coin still only halves the probability, so the cancellation is not perfect and each level is worth more than the last. Chance of first heads on flip n: n-1 tails then a head, probability (1/2)^n. Level by level: - Flip 1: chance 1/2, prize 3, contributes 	frac{1}{2} 	imes 3 = 1.5. - Flip 2: chance 1/4, prize 9, contributes 	frac{1}{4} 	imes 9 = 2.25. - Flip 3: chance 1/8, prize 27, contributes 	frac{1}{8} 	imes 27 = 3.375. The cap. All 3 flips tails has probability (1/2)^3 = 1/8 and pays 27, contribut…

Tripling coin prize, capped after three flips

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