Problem 44724. Let's Make A Deal: The Player's Dilemma 2

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, either immediately opens your door or opens another door, say No. 3, which might have the car or a goat. If the host opens door No. 3 and it has a goat behind it, he then offers you the choice, "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 assumptions:

1. The host can open any of the doors including that picked by the player.
2. The host can open a door to reveal either a goat or the car.
3. The host doesn't always offer the chance to switch between the originally chosen door and the closed doors.

It is also typically presumed that the car is initially hidden randomly behind the doors and that, if the player initially picks door D, then the host's choice of which 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 assumptions implies that all elements p_ij of the conditional probability matrix can be nonzero with the constraint that

p_i1 + p_i2 + p_i3 = 1    i.e. the host must always open a door to reveal the car or a goat. 

On the game show, you have initially chosen door D and the host, Monty Hall, opened door H, using the conditional probability Ph.

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

If switching option is not available (i.e. host opens a door with the car behind it) return NaN.