system
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human
Regression Toward the Mean versus Gambler's Fallacy: seriously, why don't these two conflict?

I understand both concepts very well, yet somehow I don't understand how they don't contradict one another. My understanding of the Gambler's Fallacy is that it has *nothing* to do with perspective-- just because you happen to see a coin land heads 20 times in a row doesn't impact how it will land the 21rst time.

Yet when we talk about statistical issues that come up through regression to the mean, it really seems like we are literally applying this Gambler's Fallacy. We saw a bottom or top skew on a normal distribution is likely in part due to random chance and we expect it to move toward the mean on subsequent measurements-- how is this not the same as saying we just got heads four times in a row and it's reasonable to expect that it will be more likely that we will get tails on the fifth attempt?

Somebody please help me out understanding where the difference is, my brain is going in circles.
gpt
They both say that nothing special is happening.

If you have a fair coin, and you flip twenty heads in a row then the Gambler's Fallacy assumes that something special is happening and we're "storing" tails and so we become "due" for a tails. This is not the case as a tails is 50% likely during the next toss, as it has been and as it always will be.  If you have a fair coin and you flip twenty heads, then regression towards the mean says that because nothing special is happening that we can expect the next twenty flips to look more like what we should expect. Since getting 20 heads is very unlikely, we can expect that the next twenty will not be heads.

There are some subtle difference here. One is in which way these two things talk about overcompensating. The Gambler's Fallacy says that because of the past, the distribution itself has changed in order to balance itself out. Which is ridiculous. Regression towards the mean tells us not to overcompensate in the opposite direction. If we know that the coin is fair, then a string of twenty heads does not mean that the fair coin is just cursed to always going to pop out heads, but we should expect the next twenty to not be extreme.

The other main difference between these is the random variable in question. For the Gambler's Fallacy, we're looking at what happens with a single coin flip. For Regressions towards the Mean, in this situation, the random variable in question is the result we get from twenty flips. Twenty heads in a row means nothing for the Gambler's Fallacy, because we're just looking at each coin flip in isolation and so nothing actually changes. Since Regression towards the mean looks at twenty flips at a time, twenty heads in a row  is a very, very outlying instance and so we can just expect that the next twenty flips will be less extreme because the probability of it being less extreme than an extreme case is pretty big.