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
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What Is the Famous Monty Hall Problem?
If 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.
The Math Behind the Monty Hall Problem
Since there are 3 doors, you have a 1 in 3 chance of randomly guessing the right door. (Yes, I know some of you might feel like your heartfelt gut choice isn’t actually a “random guess,” but in the end that’s exactly what it is.) What’s the probability of guessing the wrong door? Since the total probability must add to 1, the probability of an incorrect guess in which you win an awesome goat is 2/3.
To make this a little easier to think about, it’s helpful to imagine not just 3 doors in a game, but instead some huge number of doors—let’s say 100—which hide a huge number of goats but still a single car. For the case of 100 doors, you have a 1 in 100 or 1% chance of randomly guessing the door with the car. But 99 out of 100 times you are going to guess wrong.
But here’s the cool part: Monty is going to come along and open 98 doors and reveal to you the best option amongst the doors you didn’t select. In fact, he’s going to effectively reveal the location of the car to you 99% of the time. The only time he won’t is the 1% of the time your initial guess is correct.
With 100 doors, it’s clear you should always switch—assuming you want to win the car, that is. The same logic works with the original 3 door problem, it’s just a little less obvious because the numbers aren’t as big in your favor…but they are still very much in your favor!
The Power of Knowledge
That leaves us with just one little (OK, it’s actually pretty big) question: Why in the world is this problem so mind-bending? I mean, in the regular 3 door Monty Hall problem, despite everything we’ve learned and all the arguments I’ve made, it still seems like the final two doors must each have a 50-50 shot of being the one with the car. It just doesn’t seem possible that Monty’s random act of opening a door could matter.
Monty’s act of opening a door isn’t random at all.
Aha! Therein lies the key: Monty’s act of opening a door isn’t random at all. He knows exactly where the car is…so he knows exactly what he’s going to see when he opens that door. It’s true that if he were to instead randomly select from the remaining 2 doors (so that he occasionally and rather anticlimactically opened the door hiding the car), then both your door and the other remaining door have the same odds of hiding the car.
But Monty doesn’t act randomly. Instead, he introduces additional knowledge into the problem—and that additional knowledge changes everything because it removes the perceived randomness. So the real mind-bending in the Monty Hall problem comes not from the math, but from whatever psychology tends to trip humans up when it comes to perceiving the differences between random and non-random events.
All of which goes to show that in the Monty Hall problem—as in all of life—knowledge is power! Perhaps even the power to win a car…or a goat. Whichever you prefer.
Wrap Up
OK, that’s all the math we have time for today.
Please be sure to check out my book The Math Dude’s Quick and Dirty Guide to Algebra. And remember to become a fan of the Math Dude on Facebook where you’ll find lots of great math posted throughout the week. If you’re on Twitter, please follow me there, too.
Until next time, this is Jason Marshall with The Math Dude’s Quick and Dirty Tips to Make Math Easier. Thanks for reading, math fans!
opens in a new windowThree doors and opens in a new windowbaby goat images from Shutterstock.