most writers think that the new paradox can be defused, although the resolution requires concepts from mathematical economics. suppose formula_20 for all "a". it can be shown that this is possible for some probability distributions of "x" (the smaller amount of money in the two envelopes) only if formula_21. that is, only if the mean of all possible values of money in the envelopes is infinite. to see why, compare the series described above in which the probability of each "x" is 2/3 as likely as the previous "x" with one in which the probability of each "x" is only 1/3 as likely as the previous "x". when the probability of each subsequent term is greater than one-half of the probability of the term before it (and each "x" is twice that of the "x" before it) the mean is infinite, but when the probability factor is less than one-half, the mean converges. in the cases where the probability factor is less than one-half, <math>e(b|a=a) for all "a" other than the first, smallest "a", and the total expected value of switching converges to 0. in addition, if an ongoing distribution with a probability factor greater than one-half is made finite by, after any number of terms, establishing a final term with "all the remaining probability," that is, 1 minus the probability of all previous terms, the expected value of switching with respect to the probability that "a" is equal to the last, largest "a" will exactly negate the sum of the positive expected values that came before, and again the total expected value of switching drops to 0 (this is the general case of setting out an equal probability of a finite set of values in the envelopes described above). thus, the only distributions that seem to point to a positive expected value for switching are those in which formula_21. averaging over "a", it follows that formula_23 (because "a" and "b" have identical probability distributions, by symmetry, and both "a" and "b" are greater than or equal to "x").