154.Doubling on a Head Streak
★★★expected valueJane Street
A fair coin is flipped four times. If the first flip is heads you win , and each further consecutive head doubles your winnings. The first tail ends the run, and if the very first flip is tails you win nothing. What is the expected payout?
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The payout depends only on how long the opening run of heads is. Note that a run of length k pays a doubling amount but occurs with a halving probability.
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