X is usable. We never know how much is in the envelope, and it doesn't matter. Whatever envelope we pick up, IT WILL ALWAYS HAVE $X. The other Envelope will always have either 2x or .5x. IF you give X a value, then you are limiting your curve to only a few set of variables (mainly 1 value), and that is not a good proof of the equation. Ok say you want Evelope 1 to be X and Envelope 2 to be Y.

Therefore we have 2 cases:

CASE 1:

You Open 1: Therefore you have $X and the other has $Y, but there's a relationship:

Subcase 1: Envelope 2 has 2x the money in it: 2X=Y

OR

Subcase 2: Envelope 2 has 1/2 the money in it: .5X =Y

Therefore Y is either .5X or 2X. Now that we have good values for Y in terms of X, we can get rid of Y altogether.

Now you can reverse these variables to if you open the other envelope, BUT THE RELATIONSHIP NEVER CHANGES.

Now with those subcases, each has a 50% chance of occuring, so the EV math of:

.5(.5x)+.5(2x) = EV Is correct. IN this case, the EV is 1.25x. Therefore the second envelope will, on average, have 1.25x more money in it than the first envelope. The math is the easy part of this problem, and is correct.


BUT THAT'S NOT THE POINT OF THE PROBLEM!!!!!!

Given Girevik's contention that you can't open the envelope, and the proven math that the second envelope is Higher EV every time, which envelope do you pick first??? Does it even matter? If it is +EV to go for the second envelope, just throw the first one over your shoulder and go for the second one.

BUT.

Since the first envelope is still an unknown, it is then therefore +EV to go back to the first one again, after not looking at the second one. THEN, it is again +EV to go back to the second envelope, each time making a +1.25x EV move. Once you eventually pass out from lifting and dropping N unopened envelopes, you have made +1.25X ^ N profit from the first envelope. BUT you really haven't since the amounts havent changed. And since the envelopes will be unopened, you'll never know.

What's this prove?

Because the other envelope (regardless of which one you have now) is always 1.25x the one you have now, this just proves that the grass is always greener on the other side.