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Jacob Buckman's avatar

Nice post!

I have a tiny technical nitpick -- you make a subtle error when you say "by the Law of Large Numbers, since the X_n are independent, identically distributed random variables, this sum converges almost surely to n times the expected value". This is actually not true! The proper statement of the strong LLN is "this sum divided by n converges almost surely to the expected value". These two statements seem like they should be equivalent, but because of some subtleties around limits and convergence, they actually are not. It's fun to think about why!

One thing that might help show why your original statement must be false, is to note that by the Central Limit Theorem, the distribution of the random variable corresponding to the sum of the X_n will approach a normal distribution centered at n * E[X], which clearly means it doesn't converge in probability (or almost surely) to any one value.

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Sniffnoy's avatar

> But these zero-probability scenarios are still factored into expected value calculations, which allows expected value to be maximized by the all-in betting strategy.

If the set of these as a whole has probability zero, then it won't contribute to expected value. So that can't be what's going on; it has to be something else.

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