[bluecalm]

Yes, it's not a paradox it's just seductive flawed reasoning. Yes, at any point EV of picking an envelope at random is 3/4n (n being higher amount of money out of the two). It is all there is to it. The "paradox" is introduced by silent assumption that distribution of amounts put in envelopes is uniform which is impossible (because you can't pick numbers from infite set uniformly even if there was infinite amount of money in "adversary" disposal). The assumption is then used for conditional probability calculations: "if we see 10$ there is 50% chance the other envelope contains 20$" - BEEP, ERROR, THINK AGAIN.

Perhaps good exercise in clear thinking but not really a paradox. Good analogy is this: "If we pick random building and climb to the roof of it there is 50% chance first building we see is higher than the one we just climbed". This is obviously true, now following "paradoxical" reasoning we get: "If we climb a building randomly and see it's the Empire State Building there is still 50% chance first building we see will be higher".

This is exact analogy to reasoning about 2 envelopes problem which is supposed to lead to a paradox.

[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.

http://terrytao.wordpress.com/career-advice/there%E2%80%99s-...

[thatthatis]

Code that shows ev stays at 3x/2 (if x is the lower amount) or 3n/4 (if n is the higher amount)

https://gist.github.com/tedtieken/6567112