A statement of the problem was published in Marilyn vos Savant's "Ask Marilyn" column in Parade magazine in 1990:
Suppose you're on a game show, and you're given the choice of three doors: Behind one door is a car; behind the others, goats. You pick a door, say No. 1, and the host, who knows what's behind the doors, opens another door, say No. 3, which has a goat. He then says to you, "Do you want to pick door No. 2?" Is it to your advantage to switch your choice?
In this version of the problem, some details are not given but in the most common interpretation of the problem, and the only interpretation of interest here:
- Monty will always open another door and give the contestant an opportunity to switch.
- Monty knows what is behind the doors and will always open a door that reveals a goat.
- If Monty has a choice of goats he chooses randomly between them
- You want to win the car
A fully unambiguous, mathematically explicit version of this interpretation is:
- Suppose you're on a game show and you're given the choice of three doors [and will win what is behind the chosen door]. Behind one door is a car; behind the others, goats. The car and the goats were placed randomly behind the doors before the show. The rules of the game show are as follows: After you have chosen a door, the door remains closed for the time being. The game show host, Monty Hall, who knows what is behind the doors, now has to open one of the two remaining doors, and the door he opens must have a goat behind it. If both remaining doors have goats behind them, he chooses one [uniformly] at random. After Monty Hall opens a door with a goat, he will ask you to decide whether you want to stay with your first choice or to switch to the last remaining door. Imagine that you chose Door 1 and the host opens Door 3, which has a goat. He then asks you "Do you want to switch to Door Number 2?" Is it to your advantage to change your choice?
What is your answer to this question.