[AliasMrJones]

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If you think the independence of bets or sets of bets does not approximately hold, it might be interesting to hear why.

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Players will change their play as they win and lose. One common pattern is to loosen up when you win, but fail to tighten up when you lose. The gambler's ruin theorem means you will almost always be a big loser with this rule, even if you have positive expectation per hand. This violates independence and, with a player like this, I would base my long-term prediction of his wealth on his poor strategy rather than the mean and standard deviation of individual hands.

Another dependence is bluffing. If a player is caught in a bluff, he will be played differently on future hands. That's the point of bluffing.

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The statements above are complete claptrap. A player with a losing strategy CANNOT have long term positive expectation and a player with a winning strategy CANNOT have negative long term expectation. This is by definition. Therefore, someone who employs the strategy you outline above (loosen up when winning, but fail to tighten up when losing) will not have a winning strategy and WILL NOT have positive per hand expectation. Do you see why?

Just because a player is caught in a bluff one hand and players may play him differently on the next hand does not mean his long-term winrate and SD will change. In fact, a good player will mix bluffs into a well-rounded strategy and will be caught sometimes. This will already have been factored into the players long-term winrate and SD, assuming the sample size is large enough.

Your problem is you seem to be focused on small samples (i.e. per hand vs. long term strategy in the first part and how 1 caught bluff will affect a small number of subsequent hands.)

As I said before, the predicted results of winrate and SD with the regularly discussed formulas have been proven out through observed results of real players with sufficiently large sample sizes and posted about on this forum.