Interesting Mathematical Paradox?

[jason1990]

There are two things one could mean when speaking of the "foundations of probability." On the one hand, probability theory is a branch of mathematics and, therefore, an axiomatic system. Almost all probabilists and working statisticians use the system that results from the axioms of measure theory. Questions about the foundations of this axiomatic system, as well as questions about the relevancy of alternative axiomatic systems, are certainly relevant and important, and form the foundation of my discipline. These questions, however, have nothing to do with the envelope puzzle as it is typically presented.

On the other hand, "foundations of probability" could refer to the philosophical issues that arise when we try to interpret these mathematical statements and connect them to the real world. Strictly speaking, these questions have nothing to do with formal mathematics. A working statistician might be somewhat concerned with these issues, since he or she wants to select a mathematical model that accurately represents something in the real world. But once that model is selected, and he or she begins doing mathematics, these kinds of philosophical issues are not relevant.

The envelope puzzle, however, is not typically presented as a paradox about the interpretation of probability statements in the real world. It is presented as a mathematical paradox, similar in spirit to the puzzles one often encounters that give a "proof" that 0 = 1. To claim that it is a legitimate mathematical contradiction is absurd.

Insofar as philosophers stick to philosophy, I cannot judge whether or not they got anything wrong. But if they make mathematical claims, I have a chance of judging those. The Wikipedia article got the math wrong in at least one place. In Section 3.3.1, they say

[ QUOTE ]
But in every actual single instant when you open an envelope the conclusion is justified: you should switch!

[/ QUOTE ]
This is false. It is not justified. The conditional expectation of the contents of the other envelope, given that your envelope contains x, is not 11x/10. It is undefined, since the unconditional expectation of the contents of the other envelope is infinity.

It is natural for a non-probabilist to make this mistake. For example, a non-probabilist would likely assume it is a tautology that E[Z | Z = x] = x. (Even a probabilist might assume this for a moment.) But it is not. The statement is not true when E|Z| is infinite. In that case, the left-hand side is undefined. It should not be surprising that assuming something is true when it is actually undefined can prove a lot of absurd statements. This false assumption about conditional expectations is similar, in spirit, to assuming you can divide by zero, which is the trick by which people "prove" that 0 = 1.