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Henry Long's avatar

Your solutions to the puzzle are incorrect/poorly worded. Your answer is the payoff for the expected number of steps N. This is not the same as the expected payoff, which should be infinite. Consider a simple version, where we flip until we get a head, with the payoff as 2^N. There is a 1/2 chance that we get it after one flip, for an E[P] of $1. There is a 1/4 chance we get it after two flips, for an E[P] of $1. By induction, the payoff is infinite. For the case with 3 heads needed, the odds that we will reach N flips do not decline as N^-1, but rather as N^~(-.75), therefore meaning that the integral for this quite rapidly diverges.

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