[cabiness42]

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cabiness42: Why do you think the game will last for 2 flips? There is only a 25% chance of that result.

If you go by the most common result, that will be the 50% chance of the game lasting 1 flip, with a payout of $2. Of course, this number means nothing... suppose I offered a game with 3 equally likely outcomes: you win $1, you win $1, you win $1,000,000. I'm sure you wouldn't say this game is worth only $1 to play.

The only other way to do it (which happens to be the correct way) is to add up all the possible payouts (divided by their chances of happening, of course). The result is

(0.5 * 2) + (.25 * 4) + (.125 * 8) + ...

It never ends... it's an endless string of 2+2+2+2+2+2+2+2+2, so the result is infinity.


It shouldn't be surprising that you can't get a mathematically correct answer by going "well, half the time it's one flip, but sometimes it goes for 3 or more, so it'll probably be somewhere around 2".

That works as well as going "well, AJ doesn't win as much as AA, but it still wins sometimes, so I guess it's around 1 in 3 to beat AA".

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Well, if you read my post, I didn't just pull 2 out of the air. I calculated an expected number of flips the game would last based on the probabilities of each individual number of flips. The result was a series that converged to 2. That has absolutely nothing to do with your example of AA vs. AJ.