Probability paradox II

I had an email from Noga Alon a couple of days ago, who told me about a much better version of the paradox I discussed in an earlier post. Some of the comments relating to that post also allude to this better version. The reason it is better is that one can no longer object to it on the grounds that it assumes the existence of a probability distribution with impossible properties.

Removing the magic from Stirling’s formula

Another topic on the syllabus for the probability course I am giving is Stirling’s formula. This was lectured to me when I was an undergraduate but I had long since forgotten the proof completely. In fact, I’d even forgotten the precise statement, so I had some mugging up to do. It turned out to be hard to find a proof that didn’t look magic: the arguments had a “consider the following formula, do such-and-such, observe that this is less than that, bingo it drops out” sort of flavour. I find magic proofs impossible to remember, so I was forced to think a little. As a result, I came up with an argument that was mostly fully motivated, but there is one part that I still find mysterious. On the other hand, it looks so natural that I’m sure somebody can help me find a good explanation for why it works. When I say “I came up with an argument” what I mean is that I came up with a way of presenting an existing argument that doesn’t involve the pulling of rabbits out of hats, except in the place I’m about to discuss.

A paradox in probability

This is the first of what I hope will be several posts related to the course I am giving this term on probability.

The following is a well-known paradox. You are presented with two envelopes and told that one contains a sum of money and the other contains twice as much. You are invited to choose an envelope but are not told which is which. You choose an envelope, and are then given the chance to change your mind if you want to. Should you?

One argument says that it cannot possibly make any difference to the expected outcome, since either way your expected gain will be the average of the amounts in the two envelopes (so the expected change by switching is zero). But there is another argument that goes as follows. Suppose that the amount of money in the envelope you first choose is . Then the other envelope has a 50% chance of containing and a 50% chance of containing , so your expectation if you switch is —so you should switch.

I tried this out for real in my first lecture, and the student who was given the choice decided to switch. Rather irritatingly, he got more money as a result. Of course, the second argument is incorrect, but the reasons are somewhat subtle. My purpose in putting up a post about it is not so much to invite solutions to the paradox as to see whether it prompts anyone to give me their favourite probabilistic paradoxes. (I’ve just done Simpson’s paradox, so that one wouldn’t be new.)

General news—February 2008

Not entirely surprisingly, my hibernation is going to go on for longer than I had hoped. The reason for this is quite simple: there is a major push to get the Princeton Companion to Mathematics finished within the next couple of months or so, and if I spend time blogging then it won’t happen. There has also been a delay with the Tricks Wiki, but that may be less severe because I am not the main bottleneck for that (though there are a few things I need to do before it can be up and running). This term I am giving a first course on probability. I had planned a few blog entries on that too, and I hope I’ll have time for some in due course.