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

If you flip a coin three times, the chance of getting three heads in a row is 1/8, which makes sense because there are eight possible outcomes and they’re all equally likely.

The eight possible outcomes are
HHH
HHT
HTH
HTT
THH
THT
TTH
TTT

If after the second flip you’ve gotten two heads, then you eliminate all but the outcomes that had a tails in one of the first two spots.
HHH
HHT
~~HTH
HTT
THH
THT
TTH
TTT~~

There are only two possible outcomes remaining and they’re still equally probable, so there’s a 50% chance of getting HHH, and 50% chance of getting HHT.

The same logic works for 25 coin flips. All of the millions of possible outcomes with a tails in one of the first 24 flips have already been eliminated, leaving only two possible outcomes left.