Re: is the EV for this game really infinity?
Your solution does not incorporate any information concerning what the payoff is when the game lasts 1 flip or if the game lasts 3 or more flips as if this had no impact.
You can't just separate out the payouts, calculate the average length of a game and then later pick out the payout that corresponds to the average length of a game and ignore the rest. This isn't kosher mathematically and intuitively, the payout scheme ought to have an impact. Yes?
So your method of solution can't possibly be correct unless you were to show that indeed the longer or shorter games have no impact as in they cancel each other out or some such.
Not only do you not do this, I don't think you can do this, so your solution is incorrect for at least this reason.
Keep trying.
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