Problem 44723. Let's Make A Deal: The Player's Dilemma

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?

We will now play the game using the standard assumptions:

1. The host must always open a door that was not picked by the contestant.
2. The host must always open a door to reveal a goat and never the car.
3. The host must always offer the chance to switch between the originally chosen door and the remaining closed door.

It is also typically presumed that the car is initially hidden randomly behind the doors and that, if the player initially picks door 1, then the host's choice of which goat-hiding door to open is represented by a conditional probability matrix Ph

   Ph = [ p_11   p_12   p_13
          p_21   p_22   p_23
          p_31   p_32   p_33 ]

In the above matrix, p_ij represents the probability that the host opens door j given that the car is behind door i.

Interpreting the matrix in terms of the standard assumptions implies

 p_i1 = 0            i.e.  the host cannot open door 1, the player's initial choice.  

 p_i2 + pi3 = 1      i.e. the host must always open a door, 2 or 3, not initially picked by the player.

 p_ii = 0            i.e. the host must always open a door to reveal a goat and never the car.

On the game show, you have initially chosen door 1 and the host, Monty Hall, opened door H (2 or 3), using the conditional probability Ph.

What is the probability Pws that you will win the car by switching your choice to the door remaining?