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

I think this has been answered pretty well by others, but if any visual learners are having trouble grasping the concept, I present to you this:

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Imagine a [Binary Tree](https://en.wikipedia.org/wiki/Binary_tree). Each node is our coin at any given flip, and the two branches from each node represent heads and tails. Flipping our coin at any given node, there is a 50% chance of heads or tails, since there are only two possible outcomes. But ending up at a specific node (following the exact path of heads and tails) has a much smaller chance of happening the more we flip our coin. But remember, no matter how small the odds were of getting to a node, there are still only two equally likely possibilities forward – heads or tails.

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So yeah, if we had to bet on landing on a specific node 25 flips later on the tree only knowing the root, our chances of guessing correctly would suck. But if we have to bet on which node we hit next, we only have two options, so our chances are 50/50.