How is the gambler’s fallacy not a logical paradox? A flipped coin coming up heads 25 times in a row has odds in the millions, but if you flip heads 24 times in a row, the 25th flip still has odds of exactly 0.5 heads. Isn’t there something logically weird about that?

Question

I know it’s true, it’s just something that seems hard to wrap my head around. How is this not a logical paradox?

In: Mathematics

Answer 1

My brain won’t fully accept it either (if you really want your brain to hurt, look up the Monty Hall problem) but here’s the way I see it. There are 2 separate math problems here. The first one is “what are the odds of getting heads 25 times in a row” vs “what are the odds of getting heads just this one time”. You could then add 1 more flip and calculate the odds of getting HH. Then add 1 more flip, calculate the odds for HHH, and so forth. You’ll see the odds go way down pretty soon.

I don’t know whether this hurts or helps you, but imagine that you want the results of 25 coin flips to be HHHTTHTHTTHHHTTTHHTHTTHHT. You know the outcome can’t be a 50/50 chance. That’s a very specific pattern. And in fact has the exact same chance of happening as HHHHHHHHHHHHHHHHHHHHHHHHH.

Please let me know if this doesn’t help, so I never suggest it again, lol.