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 key is to realise that the sequence “TTTTTH” has the same probability of occurring as “TTTTTT”, and so does “THTHTH”, “TTTHHH”, “TTHHTT” and indeed any set sequence of 6 flips, which are all 1/64. Just because you’re “looking for” or placing some special significance on 6 tails in a row doesn’t actually make that sequence special at all.
In other words you can’t think “oh wow 6 tails in a row is really unlikely”, because “5 tails and a heads” is just as unlikely (1/64), it just doesn’t *seem* as “special”.
So when you’ve rolled 5 tails already, that result had a 1/32 chance. When you flip the next coin it’s a 50/50 which halves the chance of the total sequence to 1/64, then the next roll halves the chance of that total sequence again (or you could view it as doubling the number of possibilities for that number of flips) to 1/128, and so on.
You could flip TTHTHH, you could look at it and say “that’s just a random jumble, meh”, and then flip another 6 times and get TTTTTT and say “oh wow! That’s amazing! How unlikely!”… and if you wanted to see TTTTTT again you’d have to roll a set of 6 another 64 times to see it. But if you were to set out to get TTHTHH again, you’d have to try an average of 64 times to get it again, too.
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