Flipping 5 tails in a row, it’s still a 50/50 chance to get a tail in the next flip?

Question

Flipping a tail is a 1/2 chance, but flipping 6 tails in a row is a 1/64, so if after flipping 5 tails, why is it incorrect to say that your chance of flipping another tail is now lower, like you’re “bound” to get a head? I know this is the gambler’s fallacy, but why is it a fallacy? I get that each coin flip is independent, but it feels right (as fallacies often do) that in consecutive flips the previous events matter? Please, help me see it in a different way.

In: Mathematics

Answer 1

As everyone has said, the previous flips don’t matter. For me it’s easier to think of it this way though; the possibility of flipping TTTTT is the exact same as TTTTH or HHHHH or HTHHT or TTTHT. Every possible 5 combination has the exact same odds, but you’re only going to get 1 of those combinations. No one combo has any higher possibility than another, every combo has a 3.125% of occurring.