If given the cost to play, and the average win. Can I calculate the edge? (probability of winning)
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It should intuitively feel like it's not (theoretically) possible -- the cost to play and the average win are determined by a bookie (or a casino, etc.), whereas the probability of winning is determined by the game itself. Given two games, one with a probability of winning $p_1$ and the other with a probability of winning $p_2$, the bookie can adjust the returns on bets so as the cost of play and average win remain constant. For example:
In both cases the cost to play is 1 dollar and in both cases the expected win is 0, but the probability of winning is different (1/2 vs. 1/6).
In practice, however, you might be able to infer an approximate probability of winning based on past experiences of the bookie, familiarity with the game being played, etc.