How a Seemingly Simple Riddle Creates a Conflict Between Intuition And Reason

The Monty Hall Problem is a probability puzzle which troubles the mind of math students for decades. In this post I am going to share the story of how I came across this problem, how it puzzled my mind for years and how eventually I found a satisfactory explanation. Let the story begin!

Summer Vacation — Pool, University and Goats 🏊‍🎓🐐

My aunt is a statistician. Back in 1997, I was a teenager and was on summer vacation. I spent some of the vacation days at my aunt’s work place — the Tel Aviv University. My mornings would start with a one hour swim in the university pool, then continue in front of the X-Terminal at my aunt’s office, logged in to the university’s Unix servers and exploring the internet with Netscape. I’d also join her and colleagues for a lunch, appreciating the Salmon they served in the university’s cafeteria.

One day, one of her colleagues, a statistics professor who joined us for lunch, presented me with a riddle: Suppose you were in a TV show, in front of three doors. Behind one door was a shiny new car, and the remaining two doors had a goat hiding behind. So one car, two goats. You choose a door, and immediately the host opens one of the remaining doors, where you see a goat. The host then asks you whether you’d like to stay with your original choice or switch. What would you do?

Intuition v.s. Science 🔬

I immediately came up with an answer — since I now have two doors, one with a goat and one with a car, the chances are 50–50, so I would stay with the door I chose.

My young mind refused to accept the professor’s answer, who said that by switching I actually improved my chances of picking the correct door. He tried to explain in reasoning, but it didn’t click. It seemed very obvious for me that both remaining doors had the same chance for a car.

However, I couldn’t let go until I knew the truth. As soon as the lunch ended, I rushed to my aunt’s office and wrote a quick simulation in Turbo Pascal, my favorite programming language at the time. The simulation simply ran this experiment for 100 times, randomly choosing a winning door each time, then randomly “guessing” one of the doors, opening one of the remaining doors which contained a goat (like the host would do), and then counting how many times switching would lead to a win.

The program was completed and I was ready to run the simulation, hoping to see the value 50 printed to the screen. That would be prove that my answer was the correct one, and that switching the door didn’t have a better chance of winning the car. But as I translating the problem into a computer program, a slight doubt started sneaking into my head — some new way of looking at the problem began to form in my mind.

I hit Ctrl+F9. After a moment of expectation, the result appeared on the screen. It was 64. Perhaps just a coincidence? I ran it again. 67. Again. 65. It was clear —the professor was right. Switching the doors indeed had an advantage, ⅔ chance of winning. But why?

Years of Mental Dissonance

For the next couple of years, my brain lived in some kind of dissonance. It seemed like one part of it understood why switching the doors was better, but couldn’t really express it in words, while the other part thought — this is stupid, it is pretty clear that the chances are 50–50.

The Monty Hall problem came up on several different occasions, and I heard different ways to explain the mechanism behind how the statistics worked. Some people tried to explain it by presenting a variation where you had 1000 doors, with one car and 999 goats, and once you picked one, the host would open all the remaining doors expect for one (I imagine opening 998 would take a lot of time!), and after you see that all of them had goats, you’d have to make a choice whether to switch or not.

Still, you can easily look at it from the original perspective: despite the large number of doors, you eventually end up with just two doors, one has a car and one has a goat, so there chances are still 50–50. I still liked this way of looking at the problem, and it did give me a feeling that I’m coming closer to fully understanding this.

Other people tried to explain with formulas — yet, formulas can’t win intuition. It is sort of a paradox, as you need to develop some intuition for a formula in order to fully trust it. But how can you trust it if it counters your intuition at the first place?

The Resolution 🚪

As the years went by, it seemed like my mind would slowly leaned more towards switching doors, until one day I spent some time rethinking the problem and it suddenly clicked. It was suddenly obvious for me why my initial choice had a ⅓ chance even after the host opened one of the remaining doors.

The next day, I tried to explain this to my life partner Ariella Eliassaf, but despite my best efforts, I couldn’t find a satisfactory explanation. Translating my understanding into words still seemed to be lacking. Which, to me, meant I was still missing something in order to fully understand it.

After thinking about the problem for another day, I found a new way to look at it, which translated well into words, and did the trick for Ariella — for the first time, it also clicked for her. Her intuition for the problem has changed!

So here is the alternative explanation that I found:

What If You Actually Looked For a Goat? 🐐

We will start by making a simple observation: If you picked a door, and there was a car behind it, then switching would get you a goat. Similarly, if there was a goat behind the door you picked, switching would get you the car. So switching doors is actually the equivalent of negating your winning status. To put it differently, switching is like flipping a coin, where one side has a car, the other side has a goat, but you don’t know which side is up when you flip it.

Now, Imagine you actually wanted to find a door with a goat behind. By randomly picking a door, you chances would be ⅔, as two out of the three doors has goats. Using the coin analogy, the coin would fall ⅔ times with the goat side up, and only ⅓ with car side up (and goat down).

So now you can see how switching doors, which we just observed to be the equivalent of flipping the coin, also flips the chances. If you don’t flip, chances stay the same — ⅔ for goat side up, ⅓ for car side up. But once you flip the coin (switch the doors) you get ⅓ for goat side up, and ⅔ for car side up.

In other words, always switching the doors will make sure that if your initial pick was a goat, you will end up with a car, and if your initial pick was a car, you will end up with a goat.

Why Is This Problem So Confusing?

After publishing this blog post, I have spent some hours reflecting on why this problem is so confusing. So here is a quick update with my latest thoughts:

It seems like the way this problem is phrased focuses your thinking on a very specific moment, where you stand in front of two doors, one with a goat behind and the other with a car behind. If you take this narrow perspective, not considering any other information that you have, there is no reason to prefer one door over the other. They both have the same chances.

However, when speaking about probability, a single case has no meaning. The power of probability comes by looking at the information we have about the past behavior of a system, and predicting the likely behavior of the system in the future. In this case, the information we have is the rules of the game — how the host behaves, the fact that the door with the car is randomly chosen, etc.

Thus, instead of narrowing our focus to the specific moment where we stand in front of two closed doors and have to choose one (stay or switch), we should look at the entire system — the “Algorithm” of the game, and then we realize our choice is between these two alternatives:

1. Car is set behind a random door, we choose some door, host opens a door that we didn’t choose and has a goat behind, we switch a door
2. Car is set behind a random door, we choose some door, host opens a door that we didn’t choose and has a goat behind, we stay with the original door

Comparing these two alternatives is pretty complex for our brain, so I guess that it goes the easy path and narrows the situation down to “one door with goat, one door with car, we need to stay or switch”, without realizing that by taking this shortcut we lose valuable information.

Did It Click For You?

While my alternative explanation did the trick for Ariella, I realize it will probably not work for everybody. I am really interested to learn how well it works for other people, so please please please leave a comment (or write on Twitter), and tell me if it helped you to see the problem differently.

I’d also love to see alternative explanations, or if you think you can rephrase some parts of my explanation to improve it, let me know. That will make me really happy. Thank you!