Select two random integers, a and b. Clearly, the probability of b being larger than a is 100%, as the expected value of b is infinite, yet the value of a is finite. Similarly, the probability of a being larger than b is 100%, by symmetry.
What have I done wrong?
sounds like the two envelopes problem. http://en.wikipedia.org/wiki/Two_envelopes_problem (if we assume that the probability for selecting m and -m is the same, then the expectation value of b is zero. But I think that was not your intention) if we restate the problem to selecting two random non-negative integers, we can resolve it by arguing that "selecting a random integer" does not define a proper probability distribution. A possible probability distribution for selecting a random (non-negative) integer could be something like p_n= (1-1/exp(0)) exp(-n). Then the expected values for a, b are finite and identical. Is that what you meant? Christoph