Pick me, Monty!

Door Number Three
By: Jimmy Buffett, Steve Goodman - 1974

Oh I took a wrong turn, it was the right turn
My turn to have me a ball
Boys at the shop told me just where to stop
If I wanted to play for it all
I didn't know I'd find her on daytime TV
My whole world lies waiting behind door number three

I chose my apparel, I wore a beer barrel
And they rolled me to the very first row
I held a big sign that said, "Kiss me I'm a baker
And Monty I sure need the dough"
Then I grabbed that sucker by the throat until he called on me
'Cause my whole world lies waiting behind door number three

And I don't want what Jay's got on his table
Or the box Carol Merrill points to on the floor
No I'll hold out just as long as I am able
Or until I can unlock that lucky door
Well, she's no big deal to most folk
But she's everything to me
'Cause my whole world lies waiting behind door number three

Oh Monty, Monty, Monty, I am walkin' down your hall
Got beat, lost my seat, but I'm not a man to crawl
Though I didn't get rich, you son of a bitch
I'll be back just wait and see
'Cause my whole world lies waiting behind door number three
Yes my whole world lies waiting behind door number three

Jimmy Buffett knew what he was talking about back in 1974. Who didn't love "Let's Make a Deal" and Monty Hall? That show was great and it hearkens back to an era when times were much simpler.

Joe Posnanski, just about my favorite sports writer ever*, has recently re-discovered a math probability problem that references that great show. I don't know if any of you out there are math geeks. Hell, I'm not even sure I am one although I was basically a Mathematics major in college**. But I've always liked this question. Even if my heart doesn't truly believe what my brain is telling me.

Here is the gist:

You are on a game show and participating in the final showdown. You are given a choice of three doors. Behind one of the doors is a grand prize. Behind the other two doors are goats. So you pick a door, let's say Door #1. The host, Monty, knows what is behind each door. He picks one of the other two doors, a door he knows has a goat behind it. Let's say Door #3. So now he offers you the choice of keeping your original choice or switching to Door #2. What do you think? Is it to your benefit to switch or should you keep your original choice?

Most people would say it doesn't matter. There are two doors left and one of them has the grand prize. 50/50 chance. Right?

Wrong. If you switch doors you now have a 2 in 3 chance of getting the grand prize. Confused? Yeah, a lot of folks are.

The simplest way to explain it is that before Monty revealed the three doors, your chances were 1/3. Follow me? The opposite of that would be picking the other two doors collectively and your chances would be 2/3. Those probabilities don't change just because Monty revealed one of the goats. Your original choice is still 1/3, but by switching your choice you move up to 2/3. Because it really doesn't matter that he revealed one of the goats. Get it?

Yeah, I know. Doesn't seem to make sense. But it works. The catch in this question is that Monty knows where the grand prize is. He is trying to trick you by revealing one of the goats. If he never revealed anything your chance would remain 1/3. Just as if you originally only had 2 choices, your probability would be 1/2. But that isn't the case. The man is messing with you.

My heart tells me that it is two separate problems and that is where I think most folks go wrong. "I'm given a choice between three doors, my probability is 1/3. If I am then given a choice between two doors then my probability goes up to 1/2. Right?" But it isn't two separate problems. It is one problem with a devil whispering in your ear.

Don't believe me? Someone developed a simple simulation** to prove the point. Play it for a while and you will see that the chances of getting the grand prize go up drastically if you change your original pick. The heart doesn't want to believe it, but the logic behind it is true.

*His blog isn't totally about sports. You should check it out.

**I was actually a Comp Sci guy in college, but my major required almost as much Math as someone studying to be a mathematician.

***For some reason it doesn't work for me in Firefox. In fact, it locks it up. Works fine in IE, though. Dunno why. Something about the applet, I guess.

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Note: Remember to play the Bug-Eyed Trivia Challenge every day. The answer is, of course, behind door number three.