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
Assuming it’s a fair coin, it doesn’t know what happened in the earlier flips. Once you pick up the coin to flip again, you’ve removed the previous state, so the last flip can’t possibly influence the next flip. You seem to know this consciously, but it feels wrong to you because of the five in a row.
The thing about randomness is that it’s random, but your brain isn’t always good at knowing whether data looks random. If a coin strictly alternated heads and tails, that wouldn’t be random. So you have to learn to expect a certain amount of clustering. Two or three in a row should happen very frequently, and five or six in a row should happen occasionally. Even some short alternation sequences should happen from time to time. That’s part of the randomness, and your brain will see little patterns that aren’t there, and will expect them to continue.
The other thing specifically about getting multiple consecutive flips the same is that you begin to suspect the coin isn’t perfectly random. It does become more likely at some point that the coin is biased or even tails on both sides, and in those cases it would be rational to expect the pattern to continue. But that’s not randomness anymore.
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