Apr 7, 2009

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.


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


Michelle said...

Interesting reading before my run!! It is a lot to digest I am going to have to reread this. I like math problems. Good stuff green man!!!

Avitable said...

I've seen this before and I disagree with the result.

It's improper to try to work in someone's state of mind into a math equation. You can't say that Monty is trying to trick you. Math is exact and "trying to trick you" is inexact.

When there are three doors, there's a 1:3 chance that the money is behind the door you chose.

At the exact moment Monty reveals the goat, yes, this means that your chances of picking the money have changed to 2:3.

However, after he reveals the goat, the third door is now completely taken out of the equation, and your chances are 1:2. There are only two unknowns and only one right choice.

Odds are drawn and re-drawn at every given second using all available quantifiable knowledge and information. That's why I think this problem is a faulty one, because you're entering an non-quantifiable factor (he wants to trick you) in to an equation where it doesn't belong.

2abes said...

thanks, i won't get this out of my head all day, its like a bad song. i tried it and for some unexplainable reason it turns out that switching your door choice after one door is revealed does result in a better win percetange.

Verdant Earl said...

Michelle - it's a tricky one.

Adam - OK. The only reason I said that Monty is working the contestant is because the basic premise of the problem is that the host knows where the grand prize is. He has inside information. But you can't look at it as two separate problems like you are. Odds aren't really re-drawn at each step. You would agree that initially you have a 1/3 chance to pick the grand prize, right? And you would agree that would leave the remaining two doors with a 2/3 chance collectively, right? THAT'S what the problem is all about. When Monty takes away one of those doors it doesn't change the initial probabilities. Did you try the simulation? It works.

2abes - I agree. It doesn't "feel" right, but it is.

Avitable said...

Odds are always redrawn every time new information presents itself.

That's like saying if you have three horses in a race, and you bet on one, if the other horse drops out mid-race, you have a 2:3 chance of your horse winning. It's simply not true. It was true at the time the horse dropped out, but once it's gone, your chance is 1:2.

This is one of those problems I can debate to death!

Verdant Earl said...

Adam - The horse race analogy is not a good one because you would need 3 horses with exactly the same chance of winning. And you don't have someone who already knows the outcome picking which horse drops out. You can debate it to death, but you can't disagree with the results.

Ookami Snow said...

Adam - The reason the odds are 2:3 is because Monty will always reveal a goat, his pick is not random.

Think of it this way: at the start of the game you have 1:3 chance of picking correctly, or a 2:3 chance of picking incorrectly. Now when Monty opens up a goat door, you still picked yours with a 1:3 chance of being right, so now since we have more information you can change your bet to the other door which has a 2:3 chance of being right. The only way you would lose when you switched doors is if you picked the correct door in the first place.

Oh, just thought of another way to think about it. At start you have 1:3 chance of picking correctly. after revel there are still three doors to choose from, however you would never pick the goat door that was revealed. So since your original door was correct in 1:3 the last door left has to be correct 2:3... essentially if you switch you get to pick two doors, however Monty opens a boring one first for you.

We would do this problem for open house at K-State with the Statistics program. I think it made more people angry than it helped with recruiting. :)
I switched "our hands on display" to a random sampling problem when I ran the booth.

Verdant Earl said...

Ookami - there you have it. From a Statistics guy even. (I was kinda hoping you would comment)

Candy's daily Dandy said...

so you lost me at "math probability problem".

Is there a School House Rock on youtube for that??

Slyde said...

thats bunk. Yes, the new door you picked would now have a 2/3 chance of being the right door,


the door you originally picked has now had its odds improved to 2/3 as well. its all bull.

believe me. I'm in the NY times, you know....

Heff said...

These comments have my head hurting !!!

Verdant Earl said...

Candy - One would hope so, but no. :)

Slyde - what? Where the hell do you come up with that? And here I've been telling everyone you are smart. Or stupid. Now I forget which.

Heff - That was the whole point of this post. To make someone's head hurt. Congratulations! ;)

hello haha narf said...

do you know how much i drank this weekend? obviously not or you would have never done this to me. hell, my weekend hangovers didn't hurt my head this much.

Verdant Earl said...

Becky - I know how much you drank this weekend. I read about it on Twitter. Drunk Twitter girl. ;)