The Monty Hall Problem

The Monty Hall problem is an interesting exercise in conditional probability. It focuses on a 1970’s American television show called Let’s Make a Deal hosted by television personality Monty Hall.

The game would end with a contestant being shown 3 doors. Behind one of those doors, there was a prize. Behind the other 2, a goat. Monty Hall would ask the contestant to pick a door. He would then open the one of the two remaining doors showing the contestant a goat. Now with 2 doors remaining, he would ask the contestant if they wanted to change their selection. The question was – should they?

Let’s look at it. In the beginning of the game, we have 3 closed doors. We know 1 of these 3 doors has a prize behind it, the other 2 do not. So the probability of picking the right door at this point is 1/3.

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Let’s say you chose door number 1. Once you chose, Monty Hall opens door number 3 and shows you the prize is not behind that door. He then asks if you would like to choose door number 2 instead.

So what do you think the probability the prize is behind door number 2? You have 2 doors left, so you might think it would 1/2. Well, you would be wrong.

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The reality is, the probability that the prize is behind door number 2 is actually 2/3.

Don’t believe me? Don’t worry, you aren’t alone. When Marilyn vos Savant wrote about this in her newspaper column, outraged mathematics professors from all over wrote her letters protesting her faulty logic. Her response was simple, play the game and see what results you come up with.

The trick to this neat little math problem is the unique constraints we are faced with. We start with 3 doors, from which we must pick 1. Giving us a the already agreed upon probability of 1/3.

Next, one of the two remaining options is removed the equations leaving only the door you have picked and 1 other door. Common sense would tell you that you now have a 50 – 50 situation where either door gives you the same odds of winning.

But common senses fails you. If you stick to your original choice, you will have a 1/3 probability of winning… but if you change your choice to the remaining door, your probability now shifts to 2/3.

Let’s see it in action.

Below is the layout of the three possible outcomes of the game.

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For the sake of argument, let us say that we have decided to choose Door 1 to start the game.

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After we have made that choice, one of the remaining doors with a goat behind it will be revealed.

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So, now we are left with the option to stick with door 1 or try the remaining door.

If we stick to our original door, we will win 1 out of the three possible games.

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If, however, you change your door after the reveal, you will win 2 times out of the 3 possible games.

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So what is the take away here?

If you ever find yourself on Let’s Make a Deal, always change your door after the reveal.

More importantly, if you are working with data for a living, make sure you have some healthy respect for the math behind statistics and probability. The Monty Hall Problem is, at its very heart, a conditional probability problem. Issues like this and Simpons’ Paradox can lead you to develop what appears to be logically sound models that are sure to fail once they come face to face with reality.

There is a lot of great software packages out there that make building machine learning models as simple as point and click. But without someone with some healthy skepticism as well as a firm handle on the math, many of these models are sure to fail.

 

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