Monty Hall problem

Jun 15, 2014

It's easier to believe the solution by simulating it.

This is my first entry in this blog! It is rightful, as I decided to create this website after much thinking about the best way to share my Monty Hall simulation. But before showing you that, why do I need a Monty Hall simulator?

The problem

Imagine you are a player in the famous game show Let’s make a deal. You are faced with the difficult choice of picking a door out of three possibilities. I say it’s difficult because behind one door is an awesome car and behind each of the other two doors is a goat. You correctly guessed that to win the car you must choose the right door, but there is something else I haven’t told you: after you pick a door, Monty Hall, the show’s host, choose one of the remaining doors having a goat behind it and open it, giving you the opportunity to stay with your previously choosen door or to switch to the remaining one. So here is the problem:

Which choice maximizes your odds of winning?

The first time I heard the problem was in a probability class last month and I thought that I had 50% chance of winning either way. I was wrong. But I couldn’t follow as the teacher gave a wordy argument explaning why the probability of winning is 66.6% if you switch doors. A week later I started to understand the answer, thanks to this video in Numberphile’s channel:

They give a good explanation in the video, but I found myself convinced only after I watched the extended version with more math. You don’t have to watch the video or do any fancy math if you agree with the following reasoning, adapted from vos Savant:

  1. Choose a door, let’s say number one.
  2. You have a 1/3 chance that the car is behind the door. In this case, if you switch you’ll loose the car.
  3. You have a 2/3 chance that a goat is behind the door. In this case, if you switch you win (as the other goat was revealed by the host).
  4. Conclusion: if you switch, you win 2/3 of times.

It’s wonderful, isn’t it?

The simulation

I’ve also seen many people who don’t believe the answer anyway (my parents for a start), so I decided to create a simulation of the problem. To be fair, even I, after seeing all those explanations, wanted to experiment with it and “confirm” the answer was indeed correct (what a shameful aspiring mathematician!).

You can try it: Monty Hall Simulation

Written by Allan Costa

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