[jamii]

You can explicitly state the distribution and still run into the same problem: https://news.ycombinator.com/item?id=6387344 .

The underlying problem is basically that probability theory in non-finite spaces has some gotchas - one of which is that the expectation of a random variable does not always exist.

[bluecalm]

Interesting point and nice read. Still the problem is in assumption about underlying distribution of amounts in envelopes (in original case impossible uniform distribution). The reasoning is based on this assumption and leads to nonsense. What you are saying (I think) is that assuming some other distribution (possible one, instead of impossible one) could still lead to nonsense or doesn't lead anywhere at all.

[jamii]

Not so much that it leads to nonsense as that naively applying expectations doesn't always work. This is a contrived example, but it's not uncommon in eg random walk theory to hit upon cases like this where the expectation does not exist at all.

People commonly think of mathematics as being purely about formal proof but the reality is an interplay between proof and intuition. Usually when a mathematician encounters a problem in a familiar area they immediately know the answer by intuition which then guides the production of a correct proof. When you first enter a new area of mathematics your intuitions are all completely wrong and you have no idea where to start with a proof. Good teachers will introduce edge cases like this problem to refine your intuition until it is useful enough to be a guide.

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