Interesting Mathematical Paradox?

[dukemagic]

This is my first time posting in this subforum, so i apologize if I'm repeating an earlier problem, or if this belongs in probability.

My friend described this scenario to me the other day and I found it fascinating:

I have two envelopes, both of which have money inside. I tell you that one envelope has twice as much money as the other. I RANDOMLY choose one of the two envelopes and give it to you.

I then offer you the choice of changing envelopes before you open it. Is it in your advantage (EV wise) to do so? My initial though is no, because I randomly chose the envelope, and so you gain nothing by switching.

However, once you open the envelope and see how much money is inside (let's say its $20) I then offer you the chance to keep that $20 or change envelopes and take what is in the unopened one.

Do you then want to switch? The answer I arrived at is, yes, you do want to switch envelopes every time! I currently have $20. The other envelope muse have either twice as much or half as much- in other words, it must have either $40 or $10, so your EV by switching is $25 ($50 / 2).

So it seems to me that this is a paradox. There is no reason to change envelopes before you open them, but it is in your best interest to switch envelopes, after you open the first one, EVERY single time.

Anyone want to expand on this? There must be something I'm missing.