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.
Thanks for the comment. I was looking for the payoff after the expected number of steps N, but I agree that the wording could have been clearer. I'll update the solutions shortly. Appreciate you pointing that out!
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.
Thanks for the comment. I was looking for the payoff after the expected number of steps N, but I agree that the wording could have been clearer. I'll update the solutions shortly. Appreciate you pointing that out!