[pyedog]

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Now there would be a similar case for every hand and position. If one player followed these actions at a LAG table with average stats of 25% VPIP and 15% PFR for 100K hands then his personal VPIP and PFR stats would be drastically different than someone who played with these actions at a rocky table with average stats of 15% VPIP and 4% PFR.

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But over large sample size they are going to play equally much on rocky and laggy tables.

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Well that's true in general, but the table conditions still won't be completely equal for all 4 accounts.

Basically I'm refuting DWarrior's claims that they can't be bots based on minute differences in VPIP of about 0.5%. He claims that this is over 3 standard deviations which would make it mathematically improbable. However, differences of 0.5% in VPIP could be based partially on different table conditions for some of the accounts.

100K hands is a large sample size. You will probably get dealt KQo close to the expected number of times. However, when you start to break it down by KQo on the button, it becomes more specific and the sample size shrinks. Then if you break it down to KQo on the button after everyone folds, or after a 3bet, or after a bunch of callers, then it becomes so specific that the sample size is no longer that big. And this will affect the VPIP and PFR values for any bot that uses other actions to determine the best play.

Maybe I'm misunderstanding DW's calculations. But I just don't see how he could have incorporated the randomness of the other player's actions in his VPIP STD calculations. Surely this would have some kind of effect on the numbers?

We can't expect 4 bot accounts to have VPIPs of exactly 14.7326718% to within an accuracy of 10E-9 over 100K hands. Or can we?