Math help needed (EV)

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Solving numerically for a, we get roughly .3607, so the probability that you win is, to over 40 decimal places, 1.


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I can't do arithmetic before noon. a should really be about .78, which makes the probability of winning to be 1 to about 10 decimal places rather than over 40.

If your opponent only pays you $2 per hand that you win, so the bias in your favor is much smaller, then the chance of losing is somewhere in the neighborhood of 0.8%, and the game will also take several times longer to play.