**Don't be fooled by the length of this post!! It is quite possibly utter nonsense. But if you don't seek help in editing your thoughts, how will you ever learn? ðŸ™‚**

I wonder if Bayes theorem is even applicable here, maybe that's the source of the apparent paradox? To be fair I don't know enough math to propose anything definitive, but here's my chain of thought, just for teh lulz.

The simple solution proposes the following:

There are two sums, A and B. Let A be the smaller sum, and B (=2A) be the larger sum. You are presented with an envelope, and the sum inside is revealed. The sum is either the smaller sum A or the larger sum B, but you don't know which one you have. **Assuming that you switch based on the flip of a coin***, if you just so happened to have revealed the smaller sum then your expected value is (1/2)A + (1/2)2A = (3/2)A. If you just so happened to have revealed the larger sum then your expected value is (1/2)B + (1/2)B/2 = (3/4)B = (3/4)2A = (3/2)A.

*Now, I realize that the entire point of the question is to determine the proper strategy when presented with either sum. In that case, the "flip of the coin" strategy begs the question. That's a fair objection.

Having looked up Bayes theorem on Wikipedia just to try and reassure myself of the details, I came across a typical example problem. Say there is a private school with 60% boys and 40% girls, and all the boys plus half the girls wear trousers, while half the girls wear skirts. You see one of the students from a distance, and realize that they're wearing trousers. What is the probability that the student is a girl?

Bayes theorem provides a clear answer. If A = "the student is a girl", and B = "the student wears trousers", then P(A|B) = P(B|A)*P(A)/P(B) = 0.5*0.4/0.8 = 0.25. But what really happened here? You knew some information about the students initially, including how what percentage were girls and boys, and how many of those wear trousers. If you draw a Venn diagram of the situation, say with labels "Bt" for boys wearing trousers, and "Gt" and "Gs" for girls wearing trousers and skirts, respectively, it's clear to see that you've really sorted the population into subgroups along two dividing lines, gender and clothing. In one sense, Bayes theorem formalizes the process of comparing the probabilities of selecting a member from any of these subgroups by telling you which subgroups are included and which ones aren't. The basis for this discrimination is **additional information** provided by the observation of one particular attribute.

Now, what if we were to arrange an unknown number of tiddleywinks in a row, and pick one randomly while blindfolded. The tiddleywinks were indeed in a definite order, but it's not possible to distinguish one tiddleywink from another on the basis of appearance alone. However, we are asked "given that you picked your particular tiddleywink, what is the probability that the next tiddleywink you pick was laying to the right of your original tiddleywink?". The fact that you have selected a particular tiddleywink, but one which you can't identify, gives you no help in determining either the number of tiddleywinks nor the relative left/right distribution of tiddleywinkdom relative to your tiddleywink. Is the expected value of the left/rightedness of the next tiddleywink a paradox? Possibly, but it really seems to be a case of non-information masquerading as information.

What does this have to do with the problem at hand? Well, I think that the "information" provided by opening the envelope isn't really useful information at all (unless you assume a finite limit on the possible amount of money in the envelope, which is really a practical consideration and not really relevant to a purely conceptual arrangement). It really just reveals that indeed you have selected a sum somewhere on the continuum of sums from the lowest possible sum to the highest possible sum (if there even is a highest possible sum). If that's the case, then what relevance does it have to the problem at hand? I think the answer is "none at all". If that's the case, on what basis do we make our Bayesian discrimination?