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

Let’s simplify this a bit and flip 3 coins in a row instead of 6 coins in a row.

There are 8 different possible outcomes (H for heads, T for tails):

HHH HHT HTH HTT THH THT TTH TTT

So let’s say I flipped 2 tails in a row. Which means that after my next coin flip, the results are going to be either TTH or TTT. Either one of these sequences has a 1 in 8 chance of happening when flipping a coin 3 times in a row. In other words, you are exactly as likely to flip 2 tails in a row followed by a heads, as you are 3 tails in a row. That’s why the last coin flip is still a 50/50 chance.

So going back to your example, the chance of flipping 6 tails in a row is 1/64, but the chance of flipping 5 tails in a row followed by a heads is also 1/64.