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Interesting Paradoxes: The Monty Hall Problem

Mohammad Yasir is a physics graduate from the University of Delhi and currently enrolled in the master's programme at IIT.

To switch or not to switch? Amy Santiago demonstrates the Monty Hall Problem to a skeptical Captain Holt on "Brooklyn Nine-Nine."

To switch or not to switch? Amy Santiago demonstrates the Monty Hall Problem to a skeptical Captain Holt on "Brooklyn Nine-Nine."

To switch or not to switch: That's the Monty Hall Problem. This famous paradox was first posed in 1975 in a letter by famous statistician Steve Selvin of the University of California. Here is an excerpt of his version. His correspondence to the American Statistician solved the problem as well, though his explanation was widely disputed.

The problem was subsequently presented in various forms over the years, and the most recent iteration I know of is in Brooklyn Nine-Nine, where it causes a rift between our favorite Police Captain and our beloved Classics Professor of Columbia University. Here is how it goes:

The Monty Hall Problem

Suppose you're on a game show, and you're given the choice of three doors: Behind one door is a car; behind the others, goats. You pick a door, say No. 1, and the host, who knows what's behind the doors, opens another door, say No. 3, which has a goat. He then says to you, "Do you want to pick door No. 2?" Is it to your advantage to switch your choice?

Where's the Paradox?

It would be prudent to mention that the Monty Hall Problem is a special kind of paradox known as a veridical paradox. In simple terms, a veridical paradox produces results that seem preposterous but can be shown to be true quite easily.

If you recall your lessons about probability, you know that initially, with two goats and one car, your probability of getting it right is ⅓. And if you decide to switch, there are now only two doors in the picture. You either get a goat or a car. Obviously, you're not dumb enough to choose the door which the show host just opened and thus, the second goat is out of the picture. Thus, you should now have a 50-50 chance of winning, right?

Wrong!

Marilyn vos Savant, an American author known to have one of the highest IQs in the world, demonstrated in her column that if you switch doors, the probability of your victory increases to ⅔. And that is where all hell broke loose!

The Monty Hall Problem Invites Controversy

When vos Savant argued that it would be better for you to switch, more than ten thousand readers responded saying that she was wrong. And about a thousand of these readers had doctorates in various fields!

And they are not to blame. This is exactly what a veridical paradox does. Its results are absurd but nonetheless true. At first glance, a ⅔ probability seems impossible. After all, there are only two doors. The basic definition of probability states that you divide the number of favorable outcomes by the total number of outcomes. So where does the additional little bit of advantage enter the scene from?

In all possible configurations of the game, the player wins two times out of three by switching doors.

In all possible configurations of the game, the player wins two times out of three by switching doors.

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The Resolution Is Trivial

The above image clearly demonstrates that switching leads to victory in two out of three configurations. However, it still doesn't resolve the mathematical inaccuracy we just encountered. Why ⅔ when there are only two doors in the picture?

The answer lies in the way this problem is framed. Try playing the game show scene in your head. There are two goats, say A and B. If behind your initial choice of door is goat A, then the host shows you goat B, and you win by switching. If your initial choice of door leads to goat B, then the host opens the door to goat A, and you still win by switching. However, if your initial choice was the correct one, then you lose if you switch doors. Thus, switching gives you a win two out of three times.

In this way, the only thing that matters is whether you first chose the right door or not. The host showing you a goat does not change the initial probability of picking a goat (⅔) or a car (⅓).

Wait! I Still Don't Get It

Don't worry. This paradox perplexes even the most advanced minds. Let me put the answer in a different way by drawing up a table:

Note that you have picked door 1 in this scenario.

DOOR 1DOOR 2DOOR 3RESULT IF YOU STAYRESULT IF YOU SWITCH

Goat

Goat

Car

Lose

Win

Goat

Car

Goat

Lose

Win

Car

Goat

Goat

Win

Lose

Clearly, you win two out of three times by switching and thus, we have our answer. The number of favorable outcomes after switching is two, and there are three possible outcomes or configurations. Therefore, the probability is ⅔.

If You're on a Game Show, Choose to Switch!

Hence, you now know how the Monty Hall problem works and if you get to be on a game show that offers you this choice, I hope you will remember this article and switch. I also hope you're lucky enough to win after switching. Good luck and adios for now.

More Paradoxes

If you've been following me, you know all about my fascination with paradoxes and have probably read my previous articles about them. If not, there's no time like the present. Let me attach the links right here:

© 2022 Mohammad Yasir

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