Two Envelope Paradox solution with minimal mathematics

 

If the maximum possible value is finite things balance out because when one has the maximum value, swapping from it to the other will wipe out the gains in going from a smaller to a larger value. If there is no upper bound then the probability of large values must fall off, hence one cannot say that the chance of getting double the value is 50%. That only leaves the pathological cases, where although the probability of high values decreases, it decreases more slowly than do the values themselves. In such distributions the expected gain for swapping with the other value in your pair is positive in every case.

 

In infinite sets, such as the counting numbers, the mean value is infinite, whereas every actual number is finite. So every counting number is below average. Thus the mean value, which is used in gain calculations, does not indicate the expected value in the way that it does in finite sets. Because every actual value is smaller than the infinite mean, the expected gain, which is based on the mean, does not determine utility. The positive gain calculated in problematic 2EP distributions results from the mean making the grass look taller in a paddock just like ours. Thus 2EP is a disguised version of the counter-intuitive fact that every element of an unbounded set is below average.

 

Whether we open an envelope or not, we should be indifferent to switching because of symmetry: nothing suggests that one envelope is preferable to the other.