# Monty's At Random Hall

There are three doors. Behind one of them is a car. You want that car. Behind the other two doors are goats.

You pick a door, but it is not opened.

The host, Monty, opens one other door at random, and he asks you if you want to keep your selection, or switch to the third door, which has not been opened.

Monty has no idea what is behind each door, and nor do you. Is it better for you to switch your selection, or not?

• In the version of this I've heard before, he does know what is behind the doors, and the opened one is a goat. – Herb Wolfe Sep 14 '16 at 1:19
• @HerbWolfe Yes, that's the classical Monty Hall problem. – Rand al'Thor Sep 14 '16 at 1:20
• Yeah, not sure why this got downvotes. – Ben Aaronson Sep 14 '16 at 2:20
• @Deusovi No more so than the original Monty Hall, which is one of the most famous probability riddles going. – Ben Aaronson Sep 14 '16 at 2:30

There is no difference between sticking and switching.

(Doors 2 and 3 look the same.)

• Mine is based on an incorrect assumption about the question, and is not correct. – Herb Wolfe Sep 14 '16 at 2:24
• @TheBitByte "Can't happen" is for Monty opening the same door as you. "Doesn't matter" is for Monty opening the door with the car. – Ben Aaronson Sep 14 '16 at 8:46

If Monty opens the door with a car, you would switch your selection to the door Monty opened at random. If he opened a door with a goat, then there is a goat and a car left, with 50/50 probability of either being your original selection. So it is better to switch your selection if he shows you the car, otherwise it makes no difference.

In the original version of the puzzle, you have a 1 in 3 chance of getting the car if you don't switch, but now, in this version, you have a 1 in 2 chance of getting the car if you do switch, and the same chance if you don't, so switching or not makes no difference.

The reasoning is based on:

In the original Monty Hall, the contestant has twice as many (2/3) chances of getting the car if he switches, because Monty does not reveal a car.
In this version, there is a 1/3 chance of Monty revealing a car. 2/3 of the time the contestant doesn't pick the car, and 1/2 of those times, Monty will. (2/3) * (1/2) = 1/3.
The remaining 1/3 of the time, neither the contestant, nor Monty have picked the car.
1/2 of the time that Monty does not reveal the car, the contestant will pick it by switching, so, NO, it is not better to switch.
The times that Monty picks the car, switching is irrelevant, as there is no chance of picking the car.
Therefore, your chances of winning are the same whether you switch or stay.

• There's two doors left, the one you picked and the one you can switch to. If the one you can switch to has a 1/2 chance of being correct, then shouldn't the one you picked have 1/2 as well? Since the probabilities have to sum to 1 – Ben Aaronson Sep 14 '16 at 2:11
• Initially, no, as there are 3 doors to start. – Herb Wolfe Sep 14 '16 at 2:28
• Yeah, I mean after the door is opened. – Ben Aaronson Sep 14 '16 at 2:29