In a different version of the Monty Hall problem, two players are playing the game; player one chooses a door and then player two another. If the car is behind the door that no one chose, Monty eliminates one of the players at random. If one has chosen the car, Monty eliminates the other player. The survivor knows that the other was eliminated, but not the reason. Should the survivor switch?
There are multiple ways of understanding this probability. One is to follow the three possibilities
1) Player 1 selects the car, and Monty will eliminate player 2. Switching is bad
2) Player 2 selects the car, and Monty will eliminate player 1. Switching is bad.
3) None picks the car, and Monty eliminates one at random. Switching is good.
It means switching makes sense only in the case when both players select goats. The chance of this happening is one in three; this is another way of saying there is a one-in-three chance of the car not being picked by anybody (three choices are: 1 picks the car, 2 picks the car, none picks).
So here is a Monty Hall problem in which sticking to the original door is the strategy.
