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

Because you’re stipulating (and forgetting) that something improbable has already happened, with a probability of 1.

After ANY 5 random flips, the chance that you’ll complete a run of 6 tails is

* (the chance P that you’ve already flipped 5 tails) x (the chance that your next flip will be a tail). [A]

Read that bit again until you’re happy with it, because it’s the key.

Before you started, the chance that you’d flip 5 tails in a row was 1/32. The chance that you’d then flip another was, as always, 1/2. So the chance that you’d flip 6 was, as you say, 1/64.

BUT. Your flips HAVE NOW HAPPENED. We have more information. You either flipped 5 tails or you didn’t, and we know which it is. The chance P that you’ve already flipped 5 tails isn’t 1/32 – it’s either 0 or 1. The chance of your completing a run of 6 flips is either 0 x 1/2 = 0, or 1 x 1/2 = 1/2.

You’ve stipulated that you’ve already flipped 5 tails. The probability of that isn’t 1/32 any more – it’s 1. The probability of a 6th tail is 1/2.

(Try *reductio ad absurdum*. You could use the same sort of logic as you feel you want to for runs of 6 flips, for shorter runs. So how about runs of 2? The chance of two tails is 1/4, so if you’ve just had a tail, it *ought* to feel (to you at least) like a head is more likely. Similary the chance of two heads is 1/4, if you’ve just had a head, it *ought* to feel like a tail is more likely. So do it. Start flipping a coin twice, repeatedly, and record happens. It shouldn’t take you more than a few dozen trials to convince yourself that what the first flip was makes no difference.)