The problem is as follows:

Granny chooses an amount X according to probability distribution p(x) and puts $X into one envelope and $2X into another. You choose one envelope at random (with probability 50% each) and open it to find $Y. You have the opportunity to keep the amount in the envelope, or to swap and keep whatever is in the other envelope (if you trade, you will keep either $Y/2 or $2Y, depending on whether Y=2X or Y=X).

- Under what conditions should you trade, after knowing how much money your envelope contains?
- For the naive estimate of 1/2 to be correct for that probability, what would Granny's distribution p(x) have to be? Any problems with that?
- If Granny used an exponential distribution with mean $100, what strategy optimizes the expected value of the amount of money you keep?

Some hints:

+ Show Spoiler [General hint] +

The naive thinking that "it doesn't matter whether or not I should switch: I either have the envelope with X or 2X" is only valid if you don't open the envelope to find your $Y. Once you do, that argument no longer holds.

+ Show Spoiler [For first part] +

What is the conditional probability that your envelope holds $X (and hence it would be advantageous to trade), conditional on the observed value of $Y?

+ Show Spoiler [For first part] +

You don't have to find the probability for any specific distribution, you just have to find it for a

*general*distribution.+ Show Spoiler [For second part] +

You're golden once you arrive at the divergent definite integral 1/y(dy) from 0 to infinity. Hell, I think I just gave you guys the answer.

+ Show Spoiler [For third part] +

Once you have the correct inequality for the first part, the third part should just be plugging in values.

To be honest, I was surprised this question was assigned in a first probability course. Granted, the prereqs for the course are multivariate calculus and linear algebra (neither of which you have to use for this problem, though you do need basic calculus) so there aren't that many freshmen in the class, but it seemed a bit difficult for an introductory undergraduate probability course.

That being said, I was also wondering your thoughts as to the value of a basic education in statistics and probability, no matter what you decide to do in life. I find that all too often, the general public has absolutely no clue what to do when presented with probabilities or statistics, and either doesn't care at all, or arrives at completely incorrect conclusions. Do you think everyone should learn at least the fundamentals of probability and statistics?

Edited for grammar.