Let's Make A Deal

Recently I was reminded of the (in)famous Monty Hall problem. Most people that have had a probability class have heard the problem and understand the answer, but it still manages to hurt my brain.

So on "Let's Make a Deal", Monty Hall would give the player a choice of three doors, one of which had a good prize, the other two having junk. Well, the player picks one, then Monty shows a bad prize from one of the two remaining doors and asks if the player wants to switch to the last door. (This isn't exactly how the show worked, but makes the problem a _lot_ more interesting).

The question, then, is should you switch?

Now, it seems like there should be a 50/50 chance of getting a winner if you switch, because there are two doors and one winner, right? Except it's actually much more likely that the other door is the one with the prize. Bizarre.

I could go into great detail explaining how and why this works. But I won't. I'll just say that once you've seen a losing door, the door not selected has a 2/3 chance of being a winner, the door picked only a 1/3. Even though it was already known that one of the two doors (at least) was a losing door.

Now, I love math. Thinking about theoretical possibilities, ignoring real world applications, I /enjoy/ spending time that way. But that doesn't mean that Monty doesn't hurt my brain. The really important point of this problem, though, is that Monty does know which door is the winner, and doesn't reveal that one. If he just revealed a random door, it wouldn't make much of a difference. But it's hard to accept that he'd be willing or able to give /any/ help in being a winner.

While some things may seem inconsequential, you already know that one of the doors is a loser, the simple knowledge may make a huge difference.