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

By the time you’ve hit 24 heads in a row you’re already in the universe where the 1 in 16777216 chance happened, and in fact there is nothing special about it. The odds of *any* combination of heads and tails also has a 1 in 16777216 chance of happening in that exact order. You’ve thrown 24 heads in a row? Well, now there’s a 100% chance you have 24 heads in a row, you already determined that; but there sequence HTHHTHTTTTHTTTTHHHHHTHH is just as likely even if it doesn’t seem as special as 24 heads.

The 25th coin flip is simply determining if you’re entering the universe where the sequence you already have produced ends in a heads or tails. Two options: HHHHHHHHHHHHHHHHHHHHHHHH or HHHHHHHHHHHHHHHHHHHHHHHT. Those are your only two options at this point: and they are equally likely. 50:50

Edit: With the assumption the coin is fair to begin with. In the real world if someone *actually* manages to flip 24 heads in a row I’d stop thinking about the gambler’s fallacy and shift towards thinking about potential fraud.