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A fair coin is flipped until it lands heads; the game pays $\$2^{k}$ if the first head is on flip $k$ , but stops paying after flip $20$ . What is the expected payout?

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Pricing a game: EV, re-rolls and when to stop

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A fair coin is flipped until it lands heads; the game pays $2k\$2^{k}$2k if the first head is on flip kkk, but stops paying after flip 202020. What is the expected payout?

More games, decision theory and puzzles questions

A fair coin is flipped until it lands heads; the game pays $\$2^{k}$ if the first head is on flip $k$ , but stops paying after flip $20$ . What is the expected payout?