Another Interpretation of the Classic Two Envelope Paradox
The paradox could also be stated as follows. Let’s return to the statement that envelope ‘A’ has ‘$A’, and so envelope ‘B’ has an expected value of $B= ($A/2 + 2*$A)/2 = 1.25*$A. At the same time, one could say that envelope ‘B’ has ‘$B’, and therefore envelope ‘A’ has an expected value of $A= ($B/2 + 2*$B)/2 = 1.25*$B.
Your intuition says that both envelopes must have the same expected value initially. How can the expected value of $B=1.25$A be equal to $A=1.25*$B? In other words, how could 1.25*x = x?
An easy solution is to set $A = $B = zero, but that was ruled out in the initial wording of the Two Envelope problem.
The other solution is to say that $A and $B are both infinite. Since 1.25 times infinity is still infinity, there is no paradox. Galileo Galilei proved this in his “Dialogue Concerning Two New Sciences.”
Of course, you wouldn’t expect either infinite check to clear the banking system; but you wouldn’t care which check bounced, either.
Conclusions from the Two Envelope Paradox
The Two Envelope Paradox elegantly reminds mathematicians to carefully match the formulae to the experiments. In particular, the Bayesian approach requires the correct “prior” and “posterior” probabilities.
Even without Bayes, it is both possible and vital to match expectations to realities in order to solve math paradox examples such as the Two Envelope problem.
Devlin, Keith. The Two Envelopes Paradox. (2004). Devlin’s Angle in The Mathematical Association of America. Accessed July 19, 2012.