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

25 flips being heads is only unlikely because any of those 25 flips have a chance of being tails. After you’ve already flipped 24 heads, it’s no longer true that any of the 25 flips could be tails. Now there’s only one flip left that could be tails. The previous 24 flips are unable to be tails at this point. Your 25th flip can’t cause the already-set-in-stone 1st through 24th flips to change their results.

This assumes a fair coin, of course. After too many flips I would start to question the probability of living in a universe where this exact pattern happened versus living in a universe where the coin is biased and someone is cheating.