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What huge assumption? That we know nothing of the distribution, hence we must assume each distribution is as likely? You seem to fail to grasp what it means to not make assumptions.

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It makes the huge assumption that these things follow a distribution where their claims are true.

As we do not know the distribution, you can not make that assumption.

Not so - the result holds for all feasible distributions.

I just gave one where it doesn't hold.

Ah I see, apologies, I understand your concern. If we define 'big' and 'small' as being above or below the average (mean) value, then you're quite right that it breaks down for many cases, like the one you gave. But if we define 'big' and 'small' relative to the middle (median) value of the distribution, then it becomes a robust result. Hope that helps!

But if you do that, it becomes a truism, and I do not think it actually gives you any useful information any longer.

Just so we're clear, you no longer consider this to be a fallacy if by 'high' we use the median as a reference point?

"I am more likely to have been born in a high population country than a low population country"

And you don't think that's useful?

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