Archive for the ‘Probability’ Category

The Monty Hall Problem

I’m so excited. My son was demonstrating a famous┬ámath problem, and as he was going through the process, I finally understood one of the most famous questions in probability.

In short, it’s based on the old game show “Let’s Make a Deal” with Monty Hall.

Here’s the question:

You have the opportunity to win a car if you pick the correct door, #1, 2, or 3. At the beginning, what are the chances that you’ll pick the correct door at random? Keep this value in mind.

Now, Monty opens one of the losing doors at random. He then gives you a choice: keep your original door or switch. What would you do?

Most people, over 85%, stay with their original choice.

This is the wrong decision, and here’s why:

We can all agree that at the beginning of the game, the contestant has only a 1 in 3 chance of picking the correct door at random. Two-thirds of the time, h or she will pick a goat. This doesn’t change just because one of the doors has been opened. We haven’t become magically smarter just because one door was opened.

This is the key: Monty is smarter than the contestant. He has insider information and the door he opens is not opened completely at random!

Monty is in contact with the producers of the show. The producers know where the car is hidden. They won’t accidentally reveal the car. In literary terms, they’re the omniscient narrators, and Monty is their prophet.

Because Monty is a prophet, he has essentially given you an unfair advantage. When you first pick a door, you only have a 1/3 chance of winning. Monty and the producers have a 2/3 chance that you’re wrong. This doesn’t change after the first goat is revealed.

In essence, Monty is telling you, “I started with a 2/3 advantage. I am sharing this advantage with you by revealing part of my knowledge. If you switch doors, you can take advantage of my insider status and have a better chance of winning the car.”

This doesn’t guarantee you a car, but you’d be foolish not to take the 2/3 chance of winning.

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