[members_only]

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I think what you're missing is that chance of hitting one of your outs on either street = chance you hit turn + chance you DON'T hit turn AND hit river. You were just adding chance you hit turn and chance you hit river, which leads you to overestimate the probability.

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Would seem logical, but in fact it's not. This has nothing to do with it at all, read my posts again and you will understand. Working with probabilities like this one requires that you use multiplication rather than addition to get the right results.

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members_only's method is correct, and yields the same answer as your method. You can add the probabilities of events so long as they are mutually exclusive.

11/47 + 36/47*11/46 = 41.7%

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Phew, thanks! I was beginning to wonder...