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The Monty Hall Problem

The Monty Hall Problem Activity

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See more activities in: High School, Probability & Statistics

Probability problems are often among the hardest math concepts for students to wrap their heads around. Often, your common sense says one thing—but the answer is something else entirely! The only way to really hammer this concept home is through practice, practice, practice, but it doesn't have to be all work and no fun.

Here's a brainteaser that always leads to a lively debate. You’re on the game show “Let’s Make a Deal,” and Monty Hall is the host. Your job: choose one of three doors. If you choose the door hiding a car, you’ll win it! Otherwise, you win nothing.

What happens? You choose Door #1. Monty then opens the second door, and it’s empty.Your door still might be right! But then, he gives you the opportunity to switch to Door #3 before unveiling the car. Should you switch, or should you stay? The answer is surprising, and makes for a great lesson in probability. Here's how to turn this conundrum into a fun math activity your teen will enjoy.

What You Need:

  • A printout of the problem below
  • Some people to debate it (Note: Don’t look at the answer until you’re done with the debate!)

What You Do:

Read this scenario:

  1. You’re on the game show “Let’s Make A Deal” looking at 3 doors. Behind one is a brand-new car. Behind the others are gag gifts. You get to choose one door: if it’s the car, you win!
  2. You choose Door #1. But before he unveils its contents, Monty opens Door #2 and reveals a gag gift. You may have picked the right door!
  3. Monty asks if you’d like to make a deal. If you want, you can switch to Door #3.
  4. What would switching do: improve your odds, worsen your odds, or leave them the same?
  5. Debate!

The answer:

  1. When you choose Door #1, your odds of winning are clearly 1/3.
  2. Therefore, the odds that the prize is behind Door #2 or Door #3 is 2/3.
  3. When Monty reveals that there is nothing behind Door #2, it doesn’t change the original probabilities. There is still a 2/3 chance that the prize is behind door two or three. (Knowing the contents of door two doesn’t change the odds once you’ve started playing.) Therefore, door three now has a 2/3 odds of winning.
  4. You should switch!

This is a very hard concept to absorb. Our intuition tells us that once Door #2 is eliminated, Door #1 and Door #3 each have a 1/2 probability of winning. But because the odds are set for good with your first pick, the winning odds of your original door will always stay at 1/3.

I worked with a NASA scientist who wouldn’t believe this answer. So we sat down and simulated the “game” a hundred times, where I was the host and he changed doors every time. Sure enough, he won about 2/3 of the games. If you need to be convinced, try it!

Cindy Donaldson, BS Mathematics, taught Math, Business, and Computer Science at Menlo-Atherton High School for seven years. She has also worked as a tutor for SAT and SAT II test preparation. She is the mother of two young daughters.

Updated on Apr 2, 2014
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