They're not the same if the host chooses randomly. Check it out: WLOG assume the car is in door A (The rest will be the same by symmetry.) You pick A, B, or

They're not the same if the host chooses randomly. Check it out: WLOG assume the car is in door A (The rest will be the same by symmetry.) You pick A, B, or C. The host picks one of the other two doors to show you. The universes can be described by your choice, followed by the host's random choice. There are 6 possibilities: AB, AC, BA, BC, CA, CB. Since we're not in a universe where the host chose a car, we eliminate BA and CA. We are left with four possibilities: AB, AC, BC, CB. AB = You chose the correct door. (probability 1/4) AC = You chose the correct door. (probability 1/4) BC = You chose the wrong door. (probability 1/4) CB = You chose the wrong door. (probability 1/4) Since the host's choice was random, it's 50/50 as to whether the car is behind the door you chose or the remaining door. This same reasoning breaks down when the host knows what's behind the doors because you're no longer eliminating a couple of the universes and it's a 2/6 vs 4/6 situation.