The ground problem is that chance is not just something subjective. You must use the objective view, and that is that you are presented a TEP pair, and then you pick one envelope. It goes without saying that the envelope you did not choose is the one you can switch to. So you must take the pair as it is as you starting point, and not your subjective position. That is an error in the Monty Hall problem, and it is here as well. It is also the problem of kkwan. (You cannot become ‘objective rich’ by taking a ‘subjective position’). He takes the envelope that he has picked first as fix point, and that is wrong. The fix point is that there are two different amounts.
All TEP pairs can presented as (X,2X). So in general the solution is then playing through the two possible situations:
- You pick the envelope with X first. You may even look at the amount, it makes no difference. If you switch you gain X.
- You pick the envelope with 2X first. You may even look at the amount, it makes no difference. If you switch you loose X.
The subject view is that you have a certain amount, X, and assuming that the chance that the pair could have been (X,X/2) or (X,2X) are equal, you have an equal chance of finding X/2 or 2X in the other envelope. But by presenting it as such you have not used the information that you are starting by picking one amount of one pair. The two possibilities present themselves to you as
- You pick an envelope, you see the amount, X, the other contains twice as much so you gain X
- You pick an envelope, you see the amount, X, the other contains half of it so you loose X/2
So it is the same error again and again: taking the envelope of your first choice as a constant.