I think I more-or-less know what's going on here.
So let's say there's a casino that has a game where they flip 4 coins, and they only allow you to bet after a Heads is flipped. Well, if you're only allowed to play the game once, the best possible outcome you can get is to win two bets with the *exact* outcome of HTHT. In other words: H (you're allowed to bet) T (you win!) H (you're allowed to bet) T (you win!).
You can, however, lose thrice: H (you're allowed to bet) H (you lose, but at least you're allowed to bet again) H (you lose again, but at least you're allowed to bet once more) H (you lose, fyl).
But let's say you're allowed to play 6 flips: you can now get HTHTHT to win three times in a row. The only problem is that that exact combination only happens 1/64 times, whereas HHHH (wherein you lose three times in a row) happens 1/16th of the time.
However, if you're given infinite flips, you find that there are FAR more ways to win 3 straight times: it could be TTHTTTTTTTTHTTTTTTTTHT, for example. In fact, if you're given infinite flips, then there are infinite permutations where you would win 3 times in a row, and I suspect the sum of all those permutations approaches 1/16 as your sample approaches infinity.
In other words, the parameters for what samples we look at and how we analyze them is fucking with all the possible outcomes to contribute to all the various ways that bad results can follow good results.
This paper is useful in proving why certain forms of statistical analysis will give the *allusion* of the hot-hand effect. Basically, whenever we look at any finite example that necessarily excludes any case where the first several shots are misses, then we're gonna end up with more makes than misses. However, it doesn't (as far as I can tell), prove that the probability of any one flip changes based on what the previous flip was.
Reply With Quote