Amused in Review

Here's one of my favorite brain-teasers:

In the game show The Price Is Right, there always comes a moment when a contestant is offered a choice of three doors, behind one of which is a fabulous prize (a new car, a yacht, an all expenses paid vacation for two in Tahiti), while behind the other two is some booby-prize such as a year's supply of Rice-a-Roni, the San Francisco Treat.
     The play goes as follows (abstracted for the purposes of the puzzle):

• Contestant picks one of the doors
• The MC, Bob Barker (I think), says, "All right, but before we show you what you've won, let's open one of the other doors."
• The lovely assistant opens a door, behind which is...a year's supply of Rice-A-Roni.
• Bob turns to the contestant. "Now, you can have the prize behind the door that you chose, or you can switch to the other door and have the prize behind that one. Which do you want?"
• The audience goes nuts yelling advice: "Switch!" "Stay!" "Switch!" "Stay!"

According to probability theory (and assuming you don't have a Rice-A-Roni fetish) if you were the contestant, which is correct:

1. You should stay with the door you originally picked.
2. You should switch to the other unopened door.
3. It doesn't matter.

This isn't a trick question, but it is hard to see the answer until it's explained to you, and many people have trouble seeing it even after it's explained. Marilyn vos Savant, who bills herself as the smartest woman in the world, was one of the latter.

Answer, with proof, tomorrow.

UPDATE: An astute reader points out that vos Savant had it right, that you should switch. I double-checked, and my memory was faulty: it was Cecil Adams who had it wrong, at least initially (he admits he wasn't paying enough attention to the initial conditions). The moral of the story, that many bright people still have trouble seeing it, remains the same.