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July 16, 2008

Answer To “Not Metcalfe’s Paradox” Puzzle

As usual, the puzzle's been answered well by commenters. Jason points out that this is the Monty Hall Paradox and – what I shoulda checked for – the puzzle and answer are in wikipedia.

Here's the puzzle for those who missed it:

You are a guest on a game show. There are three closed doors; behind one of them is a car you want to own; behind the other two are goats you don't want despite the fact that they don't burn gas.

You have to pick a door. After you do that, the host will pick a door behind which there is a goat (he knows what's where and has to follow the rules). You then get to decide whether you should be awarded what's behind the door you picked initially or what's behind the door that neither of you picked.

The questions are:

Does it matter which strategy you pick?

If so, which strategy is favored?

What is the quantitative advantage, if any, of the favored strategy?

For extra credit: why?

The answer:

You want to pick what's behind the door that was picked by neither you nor the host. If you stick with your first choice door, your odds of winning are 1 in 3. If you switch to the remaining door after you and the host pick, you increase your odds to 2 out of 3 of ending up with the car.

But this is counterintuitive. I had to be beaten into accepting it but it's right. Here's why:

Obviously, if you pick a door at random (you have no information so your choice IS random), your odds of picking a door with a car behind it are 1 in 3; that part's easy. If you stick with the door you picked, you will win one third of the time and lose two thirds of the time.

Now suppose you follow the switching strategy. One third of the time you will have picked the door with the car initially. In this case you'll lose, however, the other two thirds of the time you'll win. That's because you are actually getting help from the game show host when you follow this strategy and he DOES have information!

Let's look more closely at what happens when you pick a goat door initially (which you will do two thirds of the time): In that case there are two doors left, one with a goat and one with a car. The host MUST pick the door with the goat (see rules above). That leaves only the door with the car which you then get to drive home. Whenever you pick a goat door first, you WILL win with the switching strategy thanks to the host eliminates the remaining goat door. Since your odds of picking a goat door are 2 in 3, you will win two thirds of the time with the switching strategy. QED.

If you don't believe me, check many diagrams in wikipedia.

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