Here's how I finally understood it (I think): Switching is only bad if you pick correctly the first time. Since you only pick correctly 1/3 of the time, switch

Here's how I finally understood it (I think): Switching is only bad if you pick correctly the first time. Since you only pick correctly 1/3 of the time, switching must be no worse or better 2/3 of the time. Since Monty will never open the door you picked nor the door with the prize, the remaining door after he opens one is the prize if you initially picked wrong, or a goat if you initially picked right. Since you initially pick right only 1/3 of the time, you switch.