Another Explanation of the Error  Before we open any envelopes it doesnt seem unreasonable to assume that if the envelopes are said contain A and A without distinguishing which envelope contains which amount then all values of A are equally likely  because we have no better information However it does not then follow from this assumption that given A is found in the first envelope the other envelope is equally likely to contain A or A for every conceivable value of A which is something we have implied in the subsequent calculation of expected value   Why  Because for any specific amount of money from which to fill the envelopes given we have found A in the first envelope for all values of A we can always find A in the second envelope whereas it is only possible to find A in the second envelope half the time  See it like this If the specific amount of money is M say then we can find A to have any value up to M but the second envelope can only contain A for values of A up to M ie half as often  So given A was found in the first envelope the chance of finding A in the second envelope is  and the chance of having A in the second envelope is   not equal as we had assumed even though all values of A are still being assumed to be equally likely