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

When you’re looking at probabilities, you’re looking at the odds of an event happening from some initial condition. The odds of getting 25 heads in a row starting from 0 is very low. However, the odds of getting the 25th head after already getting 24 in a row is 50/50.

To put it another way, the probability of anything that actually happened is always 100%. When you’re going for the 25th flip, you always have 24 flips done, so the odds of the first 24 flips in that situation are 100%, and only the 25th flip matters.