What Is the Famous Monty Hall Problem?

Ever heard of the Monty Hall problem? If so, did the world famous brain teaser leave you flummoxed? If not, how does getting flummoxed—and then just as quickly un-flummoxed—sound? Hopefully you're up for the challenge (and the fun) because that's exactly what's in store for today!

Jason Marshall, PhD
Episode #195

Monty Hall DoorsIf you’re old enough, you might remember a game show called Let’s Make a Deal hosted by a guy named Monty Hall. The show was a bit before my time, but one of the games from the show—or at least a variant of it—has stood the test of time to become one of the most debated math brain teasers ever. In honor of the show's host, it's called the Monty Hall problem.

What is this brain teaser about? Why is it so famous and so mind-bogglingly perplexing? And how can you break through the confusion and understand how to solve the Monty Hall puzzle once and for all? Those are exactly the questions we'll be talking about today!.

What Is the Monty Hall Problem?

Here's the situation: You're standing in front of three closed doors and you're told by Monty Hall that you will win whatever is behind one of the doors. What exciting prizes might you win? Behind one of the doors is a brand new car. And behind the other two? Goats.

I don't know about you, but I actually think it'd be kind of cool to have a goat…milk, cheese, yum. But apparently people were a lot more interested in the new set of wheels. So the name of the game for most people—and the one we'll be talking about today—is to guess which door is concealing the car.

But there's a catch. Once you make your guess, Monty asks for one of the two remaining doors to be opened—and he always opens a door concealing a goat. After opening this door, he asks if you'd like to stick with your original choice or switch and choose the other still-closed door.

What should you do? Does it matter in terms of changing your odds of winning the car? Which door would you actually choose if you were on the show?

Should You Switch Doors?

If you're like most people, your instinct is probably to stick with your original choice. After all, your gut told you to go with that door in the first place, and given the 50-50 nature of the decision you're facing you may as well go with your gut, right? Right?

You should switch doors. Every time. No matter what.

Believe it or not—wrong! You should switch doors. Every time. No matter what.

I know this is totally counterintuitive, but switching doors actually doubles your chances of winning the car. Of course, you might have guessed the right door to begin with, and in that case switching will cause you to lose. But the odds are that you didn't select the right door to begin with, and therefore switching is the way to go.

Don't believe me? Feeling like switching doors can't possibly help your cause? Let's think about the math.


About the Author

Jason Marshall, PhD

Jason Marshall is the author of The Math Dude's Quick and Dirty Guide to Algebra. He provides clear explanations of math terms and principles, and his simple tricks for solving basic algebra problems will have even the most math-phobic person looking forward to working out whatever math problem comes their way.

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