The Monty Hall Problem

It's great to see pieces on familiar mathematics puzzles in the mass media, so I was pleased to see this article about the Monty Hall problem on the BBC News website.  However, I'm moved to write a little piece about this, because I think it uses a rather carelessly analogy.

A brief summary of the problem.  You're in a game show, and there are three doors; behind one of the doors is a prize, behind the other two is nothing (or possibly worse, a goat).  The procedure is as follows every time the game is played:

1. you choose a door, say number 1 (but don't see what's behind it);
2. the game show host, who knows where the prize is, opens one of the other two doors, say number 2, and reveals nothing;
3. the host gives you the opportunity to either stick with your choice or door 1, or move to door 3;
4. your choice is revealed.

The question is - should you stick, or switch, or does it make no difference?

The BBC video (with Marcus du Sautoy and Alan Davies) is very clear, and so is most of the article.  However the beginning includes a reference to Deal or No Deal: the reason I don't like this is that in Deal or No Deal the banker doesn't know where the money is.  As we will see, this point is absolutely critical to getting the right answer.