Did I win this bet?

[BruceZ]

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Did the two errors cancel out just in this case, or will they always cancel out.

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They will always cancel out, if by "always" you mean for any number of outs when computing the probability of completing a hand by the river.

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As far as how I derived it, I am proud to say that I came up with that sloppy formula on my own. I can't really say why I thought it would work.

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In other words, you could not prove it would work.