To understand the solution, consider the two scenarios. $$ $$ If you stay with your choice, you win if you correctly chose the winning door on your first pick. If you switch, you win if you initially chose a door that was empty. Since the probability of you choosing an empty door is $$ P(\text{empty}) = \frac{2}{3} $$ and the probability of you choosing the winning door is $$ P(\text{winning}) = \frac{1}{3} you should always switch in this scenario. $$