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
The fallacy is because you’re confusing the odds of a single event with the odds of a series of events. You’re right that the odds of HHHHHH (six heads in a row) is pretty low. But they’re actually exactly the same as any other specific sequence of heads and tails. HHHHHH is exactly as likely as HTHTHT, or TTTHHH, or HHHHHT. Each is just one possible sequence of six coin flips. You’re kind of “bound to” get tails eventually, because as the number of coin flips increases, the number of unique combinations grows, but there’s still just one combo that’s all heads. So as you keep flipping, the odds *of your overall streak* go down. But the odds of the next coin flip, as a unique and independent event, don’t change. How could they?
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