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
Six tails in a row is 1/64. But five tails followed by a heads is also 1/64. By the time you get your five tails, the chance of both next possible outcome are (and always have been) equal.
But the important thing is that 1/64 is only the initial probability. The probability of each outcome changes as new information comes to light (it must, since many outcomes that were originally possible now have probability of 0).
If I told you regular smokers have a 5% chance of dying to lung cancer, that doesn’t mean you can go up to someone with terminal lung cancer who is on hospice and say “don’t worry, there’s a 95% chance you’ll beat this”. That 5% figure represents all smokers – you can’t apply to any given subset of smokers as you see fit.
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