 So, in this particular expression, the choosing of the host doesn't depend on where the car is, and there's only two remaining doors once "X1" is chosen (for instance, "P(H1|X1) = 0"); and "P(Ci,Xi) = P(Ci)P(Xi)" because "Ci" and "Xi" are independent events (the player doesn't know where is the car in order to make a choice).Then, if the player initially selects door 1, and the host opens door 3, we prove that the conditional probability of winning by switching is