How is the gambler’s fallacy not a logical paradox? A flipped coin coming up heads 25 times in a row has odds in the millions, but if you flip heads 24 times in a row, the 25th flip still has odds of exactly 0.5 heads. Isn’t there something logically weird about that?

Question

I know it’s true, it’s just something that seems hard to wrap my head around. How is this not a logical paradox?

In: Mathematics

Answer 1

We’re the ones calling “25 heads in a row” something special. If your favorite pattern of choice was “3 heads, 2 tails, 5 heads, 1 tail, 1 heads, 7 tails, 1 head, 2 tails, 3 heads” then THAT would be the one seeming super unlikely.

The coin has no clue about the previous flips. It doesn’t say “oh all right I’ve been heads 24 times now, I’m tired, let’s do a tails for once”. It’s just dumb physics and every flip has the same rules. We’re the ones looking for patterns and being surprised if a pattern emerges. Every particular pattern is just as unlikely as any other particular pattern.

But one specific pattern of choice is of course much less likely than “any random old pattern”, because there are countless such random old patterns and only one “favorite” pattern. We lump the random patterns into one category and we put the special pattern alone in its own category. Then of course, “25 heads” is super unlikely compared to “anything else” because there is so much more of “anything else”.