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

The odds of flipping 4 tails and then a head is 1/64.  The odds of flipping 4 tails and then another tail is 1/64.  So, once you’ve flipped the 4 heads, odds are even either way.

But, that assumes a “fair coin.”. After, say, 100 tails in a row, you might start thinking “this isn’t actually a fair coin.”.