More on the two-envelope thing
David Chalmers (insane brilliant philosopher of mind in Austrailia) has apparently written not just one paper on the two-envelope thing, but two papers on the two envelope thing! (Cross-reference: How to make people crazy with probability.)
I have it on good authority from my Mathematician Friend (TM) that the answer is that there’s no probability distribution on the real numbers. I suspect he wanted to add “so shut up about it already!!”
Hey, your Mathematician Friend doesn’t know the right math. (Math is an enormous field, he shouldn’t be embarrassed.) I don’t know much measure theory, but it’s easy to see that the value of the uniform distribution of the reals at any given point is 1/ where is Dirac’s delta function. See http://en.wikipedia.org/wiki/Dirac_delta. Arguments that other probability distributions don’t exist over the reals ultimately degenerate into arguments that the reals don’t exist either. As a computer scientist, I’m sympathetic with this view, but as a person living in the “real” world, I don’t believe it.