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

Flipping a coin doesn’t change the coin in any way.

Assume a fair coin, which isn’t two headed and doesn’t favor one side. When you flip it there’s a 50% chance you get heads. No matter what you get, the next time you flip it it’s still a 50% chance because it’s the same coin. Nothing about it has changed.

Looking at it from a math perspective, the chance to get 25 heads in a row is 1 : 0.5^25 . But the chance to have gotten 24 heads in a row, if you already got 24 heads in a row, is 1^24, or just 1. So the chance to get that 25th heads is 1X0.5^1, or 0.5.

Probability doesn’t work backwards through time. The probability of anything that happened, having happened, is always 100% after the fact.