Under reasonable assumptions about Monty Hall's motivation, your chance of picking the car doubles when you switch.
The problem is confusing for two reasons: first, there are hidden assumptions about Monty's motivation that cloud the issue; and second, novice probability students do not see how to handle the information that the opening of the door gave them.
- 1. He randomly opens doors. 2. He always opens the door he knows contains nothing. 3. He only opens a door when the contestant has picked the car.
- 1. The probability of the car distributes evenly over both remaining doors. 2. The probability of the car in your door does not change. 3. The probability of the car in your door goes to 100%.
- 1. No improvement when switching. 2. Double your chances by switching. 3. Don't switch!
Most people think that (2) is the intended interpretation of Monty's motive. However, even given this interpretation, many people at this point still have trouble with this puzzle. Here are two other arguments that sometimes do the trick.
1. Increase the number of doors from three to 100. If there are 100 doors, and Monty shows that 98 of them are valueless, isn't it pretty clear that the chance the prize is behind the remaining door is 99/100?
2. All agree that you have a 1/3 chance of picking the car originally. That means that the other two doors represent a 2/3 chance. Now, after Monty opens the one of them that is a goat (which he must always be able to do assuming he knows which one it is), the remaining door must now represent the 2/3 chance. So you should pick it.
In the real game show the situation was much more complex. Interviews with Monty Hall indicate that he sometimes lured the contestant who had picked the car with cash incentives to switch. However, if Monty always adopted this strategy, contestants would soon learn never to switch, so one presumes that Monty offered another door even when the contestant had picked a goat. At any rate, analyzing the problem when Monty was luring people away from the car is difficult, since it requires knowing something about Monty's probability of bluffing. See the paper by Fernandez and Piron for an analysis of this game.
The original Monty Hall problem (and solution) appears to be due to Steve Selvin, and appears in American Statistician, "A Problem in Probability," Volume 29, Number 1 (February 1975), page 67. It should be of no surprise to readers of this group that he received several letters contesting the accuracy of his solution, so he responded two issues later in American Statistician, Volume 29, Number 3 (August 1975), page 134. However, the principles that underlie the problem date back at least to the fifties, and probably are timeless. See the references below for details.
These references are selected from the voluminous literature on this subject because they are current and contain bibliographies:
Leonard Gillman, "The Car and the Goats," American Mathematical Monthly, Volume 99, Number 1 (January 1992), page 3
Ed Barbeau, "The Problem of the Car and Goats," College Mathematics Journal, Volume 24, Number 2 (March 1993), page 149. Contains a list of equivalent or related problems.