• Math
• Puzzles

A man comes up to you on the street that you've never met. He generously offers to play a game with you. He says that he's going to take two envelopes and fill them with money. One of the envelopes is going to contain twice as much money as the other, but he doesn't tell you what either amount will be. He will then give you one of the two envelopes randomly with a 50/50 probability of getting either. He will let you look in it, and decide if you want to keep that money or if you want to switch envelopes and be forced to keep the amount in the envelope.

Let's say that you open the envelop and see $100. Should you stay and be happy with your envelope, or should you switch and potentially get $200 or end up with $50.

Well, half the time you have the higher envelope and half the time you have the lower envelope, so your expected value for switching is:

E = (1/2) * $200 + (1/2) * $50 = $125

Your expected value for switching is more than the $100 that you would always get by staying. So, based on the math, you should switch envelopes. Easy problem, right?

But, if you think about it, the situation makes absolutely no sense. Nothing was special about $100. For any amount X, the expected value of switching is 1.25*X, so we should conclude that we should always switch when we are given the envelope. Thus, we can mathematically switch envelopes even before we look. The paradox lies in the fact that there is a natural symmetry between the envelopes. Since we no nothing special about either one, how could it possibly be preferable to switch? They are both equally likely to be the bigger one, so we should really break even by switching. It should gain us no advantage. But the math is simple and clear. So, what's going on here.

This is known as the two-envelope paradox. It's pretty interesting, I think, and it really stumped me for a while when I was first thinking about it. The solution is somewhat non-trivial but is also enlightening. So, I encourage you to think about the problem and to see if you can figure out what's wrong here. I'll post my take on the solution later.

Incidentally, Wolfram Alpha is now online, and I'm playing with it a bit.