The Monty Hall problem - convincing a skeptic I have lost count of the number of times that I have been debating the solution of the Monty Hall problem with someone.
Recently I had a long conversation with a colleague, who didn't seem to buy into it no matter what I said, but then I came up with this argument, which seemed to stir her belief. I told her: "I don't know your birthday, but I will try to guess it. My guess is August 15. Now you remove 363 dates from the calender (ignoring leap years), but you cannot remove August 15, because I chose that, and you cannot remove your own birthday, because it represents the car. What dates are left?"
Then I switched my choice, of course, and it seemed to make an impression.
Now out of interest, what arguments have you been successful with when convincing a skeptic that you should definitely switch doors?

The Monty Hall problem: You find yourself at a game show. In front of
  you are 3 closed doors, and you have been told that behind one of the
  doors there is a fancy car, and behind the two other doors there are
  goats. You don't know which door leads to what. You get to choose one
  of the doors, and after doing this, the quiz master opens another door
  that he knows for sure does not contain the car. You then have the
  option of sticking with your original choice or to switch doors to the
  other remaining door. What should you do?

 A: My favorite way of explaining it is "when is it good to switch? When is it good to stay? What are the odds of each?". That and the million doors thing, like you said are good.
A: Plates and money.
Each take out a note and have skeptic hide them under one of three plates.
Guess a plate and have her remove one of the other two.
Now pick one of the remaining (switch choice of course): you get the money if right and she if you're wrong.
Repeat the process until she runs out of money or sees the light.
(If you are feeling generous, give her back what she lost).
A: Well you start out with a 2 in 3 chance of having picked a goat.  That doesn't change when a door is revealed.  And if you did pick a goat, then the host's only choice is to reveal the other door that doesn't have the car. That's important information, as that would mean that the remaining door must have the car.  And since you had a 2 in 3 chance of having picked a goat, switching to that door gives you a 2 in 3 chance of getting the car.  It's only 1/3 if you stay, because, as we discussed, that's all the chance you ever had for the original pick.
