Just about everyone has heard that history of coin tosses does not change the odds of the next toss (though, understandably, some have a hard time trusting it). Yet in just about any behavioral psychology book, you will find a “regression to the mean” phenomena, which with its “given an outstanding performance, chances are the next one will be much worse” seems to directly contradict the Gambler’s fallacy prescription. How could that be?
Well, regression to the mean is really just a special case (or corollary) of the Gambler’s fallacy. The confusion easily arises from looking at the wrong variable.
Gambler’s fallacy rule advises: the best prediction of the next event is the expected value, the history does not matter. Example: you have 5 tails in a row, the best prediction for the next toss is still 50/50.
Regression to the mean advises: given an expected value, if we observe an outcome that deviates from it, we should expect the next outcome to be closer to the expected value. Basically, this is a special case of the Gambler’s fallacy advice. Example: you have 5 tails in a row, the chances of that are 1/32; the best prediction for the next toss is still 50/50 (or 1/2), which is regression to the mean for the odds (1/32 -> 1/2)
Note that regression to the mean rule is usually more useful in the context of performance, where the chances are distributed more finely. Given an average skill X, if you performed at a level X+10 this time, chances are the next time you’ll perform closer to your true mean, which is still X.