That's why it's a paradox.

I have a question about the Make A Deal problem -- does the host turn over a goat card (forget for the moment that I want a goat, and already have a car) whether or not I've picked the car?

Yes. Otherwise, you'd then know that the one you picked wasn't it, and he'd have just ruled out one of the other other two, leaving you knowing that the remaining one was it.

It wasn't presented as a paradox, though. Just a problem whose solution runs counter to much intuition.

When you pick yours the first time, you've got a 1/3 chance of being right. Him telling you that one of the other two isn't the car gives you no new information about the one you picked to begin with -- you already knew that at least one of those other two wasn't a car. You've still got a 1/3 chance of being right with the one you picked. But that leaves a 2/3 chance of being wrong, and that 2/3 chance is now entirely on the still hidden unpicked one.

(I really can't figure out a good way to explain this entirely. The way I was able to totally convince myself was just to set up all the different possible situations and look at the results.)

If you can't explain it to me, Hil, I don't feel so bad about not getting it.

Let's say that the car is in door 1. If you picked door 1, then you'll get it if you stay. If you picked door 2 or door 3, then you'll get the car if you switch. There's a 1/3 chance you picked right on that first try, but a 2/3 chance you picked wrong, so in 1/3 of the cases you'll get the car if you stay, but in 2/3 you'll get it if you switch.

I'm glad others are confused because I din't get it last night.

What's at the core of the problem? Being given the chance to switch to a card that has a 1/2 chance of being the right one? I mean, 1/2 if you started from step 2?

You know, this might be one I just accept, or run the numbers on. My big issue is that I want there to be some calculation that gets you from the odds on the first pass to the new odds, and I just can't see it.

What's at the core of the problem? Being given the chance to switch to a card that has a 1/2 chance of being the right one? I mean, 1/2 if you started from step 2?

I think the core of the problem is that the odds don't change. You've got a 1/3 chance of being right, 2/3 chance of being wrong. Suppose they don't tell you "door 2 is not a car." They just say, "You can stay where you are, or you can switch to doors 2 and 3. If either of those is a car, you get it; if neither of them is, you don't." That's an equivalent problem -- knowing which door of 2 and 3 is the one without a car doesn't change anything.

Aha! All the potential lives in the one card. I'd still rather see some multiplication, but I can live with that.