# Non-Normalizable Probability Measures for Fun and Profit

Sean Carroll over at Cosmic Variance:

Here’s a fun logic puzzle (see also here; originally found here). There’s a family resemblance to the Monty Hall problem, but the basic ideas are pretty distinct.

An eccentric benefactor holds two envelopes, and explains to you that they each contain money; one has two times as much cash as the other one. You are encouraged to open one, and you find \$4,000 inside. Now your benefactor — who is a bit eccentric, remember — offers you a deal: you can either keep the \$4,000, or you can trade for the other envelope. Which do you choose?

If you’re a tiny bit mathematically inclined, but don’t think too hard about it, it’s easy to jump to the conclusion that you should definitely switch. After all, there seems to be a 50% chance that the other envelope contains \$2,000, and a 50% chance that it contains \$8,000. So your expected value from switching is the average of what you will gain — (\$2,000 + \$8,000)/2 = \$5,000 — minus the \$4,000 you lose, for a net gain of \$1,000. Pretty easy choice, right?

A moment’s reflection reveals a puzzle. The logic that convinces you to switch would have worked perfectly well no matter what had been in the first envelope you opened. But that original choice was complete arbitrary — you had an equal chance to choose either of the envelopes. So how could it always be right to switch after the choice was made, even though there is no Monty Hall figure who has given you new inside information?

Here’s where the non-normalizable measure comes in, as explained here and here. Think of it this way: imagine that we tweaked the setup by positing that one envelope had 100,000 times as much money as the other one. Then, upon opening the first one, you found \$100,000 inside. Would you be tempted to switch?