The talk about conditional probability reminded me of one of my favorite mathematical paradoxes - the so-called Monty Hall Problem. Warning: This may confuse you! Don't read any further unless you really like puzzles and paradoxes.

The Monty Hall Problem

There was once a TV game show called Let's Make A Deal which was hosted by Monty Hall. He played different games with the studio audience similar to the way Drew Carrey does in The Price is Right.

One of the games goes like this:

1. Monty presents to you three doors, labeled 1, 2 and 3. He tells you that there's a car behind one door (it's a big door) and goats behind the other two. You get to pick one door.

2. After you make your choice, Monty opens one of the other doors to reveal a goat. He then gives you the option to change your original choice to another door.

The question is: Would you switch?

I'll give you some time to ponder...

Time's up!

Intuitively, you would think that there's no real advantage to switching. At step 1, you had a 1 in 3 chance of picking the car. At step 2, you now have a 1 in 2 chance of picking the car. Your original choice is still no better than the other door. Both have a 50/50 chance.

I'm going to stop here for two reasons:

1. I have a meeting to go to.

2. I'm still not sure why the intuitive solution is incorrect.

In the meantime, I'll refer you to two web pages:

Wikipedia - Monty Hall Problem

Play the Monty Hall Problem online

Excerpt from The American Statistician

I hope to come back to this soon with some clarity.

...

Ok, I'm back with a fairly simple explanation (although not rigorous) that I think makes sense...

When you make your initial door selection, you may have chosen the car or goats:

Case 1: 1/3 chance of having chosen the car

Case 2: 2/3 chance of a goat.

In case 1, after Monty reveals a goat in another door, switching is bad since you were correct with your initial guess.

In case 2, after Monty reveals a goat, switching is good since your first guess was a goat and Monty's door was a goat and switching would get you to the car.

Therefore, there's 1/3 chance that switching is bad and 2/3 chance that switching is good. Therefore, you should switch!

Make sense?

I'd really like to work on explaining this in terms of the conditional probabilities that we learned in class.

## Tuesday, January 15, 2008

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