Hmmnyeees. Monty is constrained not only by -not- opening the door I've picked, but *also* by not opening the car door.
Monty's choice is always narrowed down to two possibilities - a random pick or an instruction. 66% of the time he gets an instruction.
Monty has no choice if "we" need (as a supervisor controlling the rules of the game) to tell him to pick door notCar(B) or notCar(C). These are equivalent (non)choices as far as the original pick is concerned.
He can pick notCar(A[1] A[2]) if I as the player pick the car.
The information he actually gets is whether my choice was the car, and if it wasn't, he gets an instruction to pick notCar(B) or notCar(C)
If I picked the car, his choice means nothing.
If I didn't pick the car, he reveals an option which is not the car, which to me is equally likely to be the car or not.
The information I get is that he's picked a door that I didn't choose which is a goat.
There has to be a very plain statistical or quantum calculation for this if it's mechanical; I can see that his choice is narrowed by me, and I can see that my second choice is constrained by him, but I'm struggling to understand the difference between Car(B) and Car(C) because in one state I choose B and in the other I choose C, and both seem quite equal.