Hold on, going through the calculations for the second case you described (the, “picks one of the other 2 doors at random” one). (1/3) chance you initially pic

Hold on, going through the calculations for the second case you described (the, “picks one of the other 2 doors at random” one). (1/3) chance you initially picked car. Then, (2/2) chance the door he reveals has a goat. (2/3) chance you initially picked a goat. (1/2) chance he reveals the car (but we will condition on this not happening) and (1/2) chance he reveals goat. Sanity check : these (1/3)(2/2) + (2/3)(1/2) + (2/3)(1/2) add up to 1, as they should . For the cases where he doesn’t reveal a car, and we should switch: (2/3)(1/2) For the cases where he doesn’t reveal a car, and we shouldn’t switch: (1/3)(2/2). So, yes, (1/3)/((1/3)+(1/3)) = 1/2 , as you say.