[BBB]

In the envelope problem, to come to any conclusion other than that of argument #2 requires you to make one or more assumptions which are not given in the original problem. If you cannot derive any useful information from viewing the content of one of the envelopes, then it doesn't matter if you switch or not. Argument #2 is flawed if and only if you somehow gain information beyond that presented in the original problem by viewing the contents of one of the envelopes.

The flaw in argument #1 is that it assumes that, regardless of the amount that we see in the envelope we open, there is a 50-50 chance that the amount in the other envelope is double this amount, as opposed to half this amount. This statement cannot possibly be true for all values that we may encounter in the envelope.

Put simply, when we see that our envelope contains $100, we know that the other envelope contains either $200 or $50. We do not know, however, the chances that it contains either amount.