Not really a paradox, but a puzzle in Probability theory.

Suppose I have 2 envelopes. I put $100 into the first, and either twice as much ($200) or half as much ($50) into the other, with equal probabilities (0.5 each). I give you the first envelope (with $100); you may either keep it or switch. What should you do?

Well, if you stick with your envelope, you always get $100. But if you switch, you get $50 with probability 0.5 and $200 with probability 0.5, so you expected profit is $125. Clearly you should switch, for an expected gain (just for switching) of $25.

That's not the paradox. This is...

Now I take my 2 envelopes; I put some random sum $N into the first, and $(2*N) into the second. I toss a coin (independently of N, of course) and give you the first envelope if it comes up heads, or the second (if it comes up tails). I let you examine the envelope, and inside you find $M (M is either N or 2*N). I offer you the possibility of switching.

You definitely want to switch! By the above logic, the other envelope contains $(M/2) with probability 0.5 and $(2*M) with probability 0.5, so if you switch you expect to gain $M/4. But the same reasoning holds even if I don't let you see the sum of money in the envelope! (it's just a random number, after all, and your decision is always to switch, no matter what).

So given the above 2 envelopes, which are indistinguishable, you always want the other envelope.

If that isn't weird enough, consider what happens after you switch: since you didn't look inside the first envelope to decide to switch, you don't need to look inside the second to decide to switch again. And every time you switch, you expect to "gain" more money (in an exponential series!).