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Comment by CJefferson | Hacker News Reader

The problem with this is very subtle.

The problem is saying that it is equally likely that the other envelope contains half, or double, the money of the envelope we have chosen.

It is impossible to choose a value from an infinite series (the list of possible monies), with every value having an equal opportunity of being chosen.

Therefore the probability of money distribution must be non-even in some way, and the argument falls to pieces.

It is not reasonable to say "Well, I don't know the probability, so I'll treat it as 50/50".

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No, that's not the problem. You don't know how much money is in either envelope, but it needn't have been selected randomly. The only randomness is whether you the envelope you choose has the greater or lesser value, which is just a uniform distribution on two points.

That being said,

>It is impossible to choose a value from an infinite series (the list of possible monies), with every value having an equal opportunity of being chosen.

this is also incorrect. You're right to say that you can't have a uniform distribution over the possible values of money, but that's not just because it's infinite. You can have a uniform distribution over [0,1] for example, which is larger than the set of possible money values.

>You don't know how much money is in either envelope, but it needn't have been selected randomly.

This is exactly wrong, at least in the mathematical formalization of this problem that everyone is assuming. The definition of "don't know how much money" must be taken to be "has a uniform distribution".

If you want to use some reasoning that treats the amount of money as an "unknown in an equation" then you are instead giving a solution that differs for each possible value of the money and does not capture the random choices we are trying to model.

By the way, no one said that uniform distributions on infinite sets don't exist. He said "infinite series (the list of possible monies)", a slightly mis-used term that clearly excludes [0,1], your chosen example.

This is precisely the correct answer — and it's disappointing to see a less good answer getting so many votes when this one hasn't had any.

The Hacker News community has a strange and sad relationship with actual math, that is, math as mathematicians define it. Math formalizable in ZFC and not arguments of A/B test effectiveness and VC funding.

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