Everybody's been exposed to statistics at some point or another. The study of statistics is two-fold. First, there's applied statistics, which we use in applications like my Variance & Bankroll Calculators, that let us know if what we're experiencing is out of the norm. Second, there's statistical theory, which can help us avoid making general errors in judgment.
Three major theoretical concepts stand out, the Law of Large Numbers, the Central Limit Theorem, and Sampling Bias.
1 - Law of Large Numbers
As our sample gets larger, all of our observed numbers get closer to their true values. OK, easy enough! But did you know, you can make "pretty good" justifications with a very small sample? For example, say you sit down at a 9-handed table with an opponent who plays very tight when in a good mood but plays very loose when tilted. If he raises his two hands in a row, you can assume with over 90% certainty that he's not playing tight. So, by the second raise, you could already be thinking about three-betting this particular opponent.Take Home Message: Some things need a very large sample, some things need a very small sample.
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2 - Central Limit Theorem
Many statistics we use in poker have a normal distribution. This means, if our sample is outside of what we expect, there's a better chance of getting what we expect in the next sample. This is NOT the "gambler's fallacy" which would erroneously state that if a coin comes tails ten times, it is more likely to come tails or heads, next. Instead, this says that if a coin comes tails seven out of ten times, it is more likely to come tails <7 out of ten times in our next sample, than it is to come tails 8 or more times out of ten.Take Home Message: If you achieve a sample ROI of 10%, it is more likely that your actual ROI is <10% than >10%, because most players ROIs are <10%.
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3 - Sampling Bias
Sampling bias occurs when we look at a non-representative sample, like a sample bounded by two events. A "500 game break-even stretch" is not a representative sample of games. First, we wouldn't even be looking at the sample if we had performed as we expected. Second, are selecting the start of the sample to be when we ran very well, and the end of the sample to be when we ran poorly.Take Home Message: It's almost impossible to get an unbiased survey in poker.
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