::::::Of course, as soon as a door plays a specific role, we have to conditon on it. As we discussed: label the doors, the simplest way is numbering them 1,2 and 3. Introduce the random variables C (car), X (first choice) and H (host), each taking the possible values 1, 2 and 3. At the moment of the decision to switch or not. we know: X=x and H=h; x<>h. Etc. Notice the difference between X and x, and between H and h. Both X and x may be described as the door first chosen by the player (and analogous H and h). Yet they are different, and in just plain text the difference is often not noticed, and hence leads to misunderstanding.