Any scientist who couldn't explain to an eight-year-old what he is doing is a charlatan.

During EIS Group architect bootcamp in Odessa in my presentation Architect Soft Skills I asked the participants (as a knowledge transfer exercise) to explain Monty Hall problem to the colleagues who don't know it.

We didn't have enough time to complete the exercise, but I am sure that I succeed in showing the most important thing, sometimes it is hard to explain even the simplest thing. And it is very important to keep the right for the partner to answer “I can’t follow you; you lost me”. The goal here is to avoid sentences like “you have to study the basics of math” or “this is obvious, please think about this when you have spare time”. The goal is to share the knowledge by sequence of the logical statements and the person who explains the things is responsible for that.

What was really funny is that I myself failed to persuade some people that my solution is correct and this is an interesting result to consider it separately.

Anyway, I promised to provide simple and logical explanation. Here it is.

The problem statement

Here is the problem statement to be on the same page:

1. We have a TV show where host and player take place.
2. The host offers to the player 3 closed doors with the prize behind one of them. The player is supposed to win the prize if he/she guesses the door behind which the prize is stored.
3. The player makes the first guess, pointing to one of the doors.
4. The host who knows where the prize is located, says “I won’t open this door right now. Instead I will open another one where there I no prize” and does so.
5. After that the host asks the player: “Would you like to change your mind?”
6. The player has two options: either to keep the choice or to change his/her mind.

The question is, what is the winning probability in case of keeping the choice or changing mind and why?

Note

The problem, indeed is counter-intuitive and most of the people choose one of the typical answers:

1. It doesn’t matter if the player changes mind or not. The probability to win with two closed doors is 50%.
2. When the played made first decision the probability of win was 33% and at the second turn it is 50%, so the player should change his/her mind.

The problem is beautiful not only because it is counter-intuitive and seems to be simple and obvious, but because it is a perfect example that for any problem there is a clean, simple and straightforward solution.

I heard many explanations, but most of them don’t sustain the question “why do you think this statement leads to that one?” and often come to the answers like “this is obvious” which most often are used by those who really unable both to truly understand and explain.

Explanation

Initially the probability to have the prize behind each of the doors is 1/3, since we have 3 doors and the probability to have the prize behind each and every of them is equal.

When the player has chosen the door we have two options who the situation could revolve:

1. The player has guessed the door with the prize correctly (the probability is 1/3)
2. The player didn’t guess the door with the prize (the probability is 2/3)

Let’s consider the situation for the host in case the player has guessed the door with the prize. In this case the host has the choice which door to open and it would be one of the resting doors. And the last door won’t have the prize behind it.

In this case if the player changes his/her mind they will lose. If they don’t change their mind, they will win. And here we don’t have probabilities. The outcome is predefined by the player’s behavior.

Please, don’t forget that we have such a situation in 1/3 of the cases.

Now let’s consider the situation for the host when player didn’t guess the door with the prize. In such a case the host don’t have a choice which door to open, he/she is obliged to open the only door which is left without the prize behind it. The host is constrained since the player already has chosen on of the doors behind which there is not prize and the host can’t open it.

The outcome is predefined again and host can’t change it. He is just a robot.

If player changes his/her mind, they win; otherwise they lose. Again, without any probabilities.

Please, don’t forget that we have such a situation in 2/3 of the cases.

The result: if the player changes his/her mind, they win in 2/3 cases.

Addition

The key here is that in this game the host doesn’t have choice. In the first case he/she has an illusion of choice, he/she could choose any door which is not chosen by the player, but this choice doesn’t matter, all the doors are equal for the game. In the second case the host doesn’t have the choice at all, he/she must open the only available door.

Feedback

If somebody sees more simple explanation for the problem or some inconsistency in my explanation, please drop me an email. The colleagues from EIS Group could use the corporate email meanwhile others could use my "in house" email.

Most interested people could write a program which simulates this game. Please share your code if you like, I will publish the best solution here. Mine took 76 lines of C++ code.