Re: buying in short
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using my standard deviation, just shy of 250k hand samples would narrow your win rate to a 3BB/100 range (95% confidence),
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Your standard deviation appears to be much higher than mine. A side benefit of buying in short is that my SD is lower.
I don't think your win rate is so much lower that you need to get the 95% confidence interval to be so small to reject the hypothesis that buying in for 50 BB is only half as profitable as buying in for 100 BB.
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With 20k hands all you can say is it's not costing you 10.75BB/100 (again using my SD, and assuming observed win rate for both is exactly the same).
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That's not true. Even if that were roughly 2 joint standard deviations (it's a lot higher than mine), you don't need 2 standard deviations of evidence before you can say anything. A Bayesian approach might say that you should reweight the hypothesis of equality upward by a factor of 3.1 relative to the hypothesis that there is a 1.5 standard deviation difference. (3.1 = exp(1.5^2/2)) So if you started with the assumption that the two were equally likely, you would update that to saying that equality is a 3.1:1 favorite over the difference in win rates that would be 1.5 standard deviations away from the observation. That's not bulletproof, but it would be a lot better than NO evidence, which is what thedustbustr offered.
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