You are obviously using 'probability' in a different sense from me. As far as I know it is a simple number, a scalar, it has no properties except a magnitude.  In the sense that I use it, the probability of winning the car in the MHP if I swap is the same as the probability of initially picking a car in a game with 2 cars and one goat, it is 2/3.  The questions may be completely different and the probabilities my refer to different concepts but they are both 2/3.  They are both have the same probability. What exactly do you mean by 'probability'?