The classic example of this is flipping 100 coins. If you get heads 99 times in a row... the last coin is still 50-50. Yes, it is obscenely unlikely to get heads 100 times in a row. But it's already obscenely unlikely to get heads 99 times in a row. And it is obscenely unlikely to alternate perfectly between heads and tails. And it is obscenely unlikely to get a binary pattern spelling out the alphabet. And it is obscenely unlikely to get... literally any pattern.

Every pattern is equally unlikely, with a fair coin. We see 99 heads in a row versus 1 tails at the end, and think it narrowly averted the least-probable outcome. But only because we lump together all sequences with exactly one tails. That's one hundred different patterns. 1-99 is not the same as 99-1. We just treat them the same because we fixate on uniformity.

Compare a non-binary choice: a ten-sided die. Thirty 1s in a row is about as unlikely as 100 heads in a row. But 1 1 1... 2 is the same as 1 1 1... 3. Getting the first 29 is pretty damn unlikely. One chance in a hundred million trillion. But the final die can land on any number 1-10. Nine of them upset the pattern our ape brains want. Wanting it doesn't make it any more likely. Or any less likely.

It would be identically unlikely for a 10-sided die to count from 1 to 10, three times in a row. All the faces appear equally. But swap any two events and suddenly it doesn't count. No pun intended.

If this couple had eight children, for some god-forsaken reason, and you saw seven boys, the eighth kid being another boy is not less likely for it. The possibility space has already been reduced to two possibilities out of... well nine, I suppose, if order doesn't matter. They could have 0-8 boys. They have at least 7. The only field that says the last kid's not a coin toss is genetics, and they say this guy's chromosome game is strong.

[–] starman2112@sh.itjust.works 6 points 1 year ago* (last edited 1 year ago)

You're right, but it's not a subversion of the Gambler's Fallacy, it's a subversion of conditional probability. A classic example is that I have two kids, and at least one of them is a boy. What is the probability that I have two boys?

The intuitive answer is 50%, because one kid's sex doesn't affect the other. But when I told you that I have two kids, there were four possibilities: GG, GB, BG, or BB. When I told you that at least one of them is a boy, all I did was take away the GG option. That means there's only a 1 in 3 chance that I have two boys.

But by having one child answer the door, I change it yet again–now we know the sex of a particular child. We know that the child who opened the door is a boy. This is now akin to saying "I have two children, and the eldest is a boy. What is the possibility that I have two boys?" It's a sneaky nerd snipe, because it targets specifically people who know enough about statistics to know what conditional probability is. It's also a dangerous nerd snipe, because it's entirely possible that my reasoning is wrong!