For those of you who are too young to remember Let's Make A Deal, the host, Monty Hall, stands in front of three closed doors. Behind these doors lie two fabulous prizes and a gag gift, such as a goat eating a copy of War and Peace. He asks you, the contestant, to select a door, then he opens another door revealing one of the two fabulous prizes. He then asks you if you want to change your mind and opt for the remaining door, which you did not originally select. Should you switch?
The answer is clear if there are 500 doors. You pick one, Monty shows you 498 fabulous prizes, and there is one last closed door to consider. You certainly didn't select the one "bad" door at the outset. If you switch now, you're sure to bring home a goat. Stick with your original choice!
The math is not as extreme with three doors, but the answer is the same. Let's say you switch. You selected a good door at the start with probability 2/3, and now you blew it! Only 1/3 of the time, when you selected the bad door, do you become a winner. Now - let's say you stand pat. You're a winner 2/3 of the time, and that's the end of it.
The situation is reversed if there is only one fabulous prize; you should switch doors if given the opportunity.