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
Probability works by multiplying independent events together (so in your example, a coin coming up tails 6 times is 1/2* 1/2…).
However, once you flip the coin, the probability that it was tails is either 1 (if it was tails) or 0 (if it was not). So we can rewrite the likelihood that we get tails 5 times given we’ve gotten tails 5 times as 1*1*1*1*1, right? It’s happened, so there’s a 100% chance it’s happened. So the probability of 6 tails given 5 tails is 1 *1/2, or 1/2.
The fallacy is thinking that the past probability still has an effect on the future outcomes; so thinking that the probability of not getting heads in 6 coin tosses is (1- probability of all tails; 1-1/64, 63/64). However, given the 5 coin tosses already happened, it should be (1-probability of tails; 1-1/2, 1/2).
In addition: you are thinking “the odds of not getting heads”. However, what you’re really determining is “odds I get 5 tails and then 1 heads” vs. “odds I get 6 tails”. Odds you get 6 tails is 1/64, odds you get 5 tails and then heads is 1/64, the odds are ultimately 1:1.
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