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 reason is because your brain notices unusual patterns better than other patterns, but mathematically they’re the same.
If you flip 6 coins and get HHHHHH, your brain notices that. You would also notice TTTTTT, and you might notice HTHTHT and HHT-HHT.
It’s easy to get a cognitive bias of, “If this coin is random, then after getting HHHHHH I’m bound to get a T soon.” Likewise, if you got HTHTHTHTHTHT, you might start to expect a double heads or double tails soon. And HHTHHTHHTHHTHHT might make you think you’ll get a THT or TT somewhere to break up that unusual pattern.
Mathematically, all of these strings are equally likely. When you flip a coin once, H and T are equally likely. When flipping twice, HH, HT, TH, and TT are all equally likely. Likewise, when flipping three times, HHH, HHT, HTH, HTT, THH, THT, TTH, TTT are all equally likely strings. The cognitive bias is your brain notices HHH as unusual but does not notice HHT as unusual, even though they have exactly the same chance of showing up.
The odds of getting HHHHHH is indeed 1/64. However, the chance of HHTTHT is also 1/64, as is TTHTHT. The issue is your brain only notices 2/64 strings containing only heads and tails, and doesn’t notice the 62/64 strings containing a mixture of heads and tails, even though the two are equally likely.
The confusion is that, with 1 flip, the odds of getting all heads or all tails are 2 strings out of 2 possibilities, or 100%. With 6 flips, the odds of getting all heads or all tails are 2 strings out of 64, or 1/32. So even though the odds of getting any given string is 1/64, the odds of getting a string that you *don’t notice* are over 95%. And that’s what causes the gambler’s fallacy: the disconnect between the actual probability, and the patterns that your brain notices.
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