Mrs. Peel has a post on Monty Haul and how it doesn't apply to Deal or No Deal. It's boring math stuff, but is just the kind of thing I like. I understand and concur exactly with her conclusions.
Now, here's the situation on DoND: you have picked a case, eliminated the rest, and now you are down to two cases. You know one has the million, and the other has one dollar. Should you switch, or stick with the case you picked?
It makes no difference what you do. Here's my thinking:
If there are 30 cases, your chance of picking the million at the outset is 1 in 30. So the chance the million is in one of the other cases is 29/30.
You eliminate a case. The chance it was the million case is 1/29. There are now 28 cases left.
In fact, there are 28*27*26... ways to eliminate the remaining cases to get to having just two cases, the 1 and the 1 million. That's a lot of potential paths. The chance that you follow one of these paths is about 28!/29!, or 1/29.
So it seems to me that when you get to two cases, there's a 1/30 chance the million is in case 1, and a 29/30*1/29 = 1/30 in case 2.
Ergo, do what you want, since the odds are even now.
I'm not enough of a math stud to be sure this logic is correct, though. Anybody else out there who is?