Whether you toss a coin to decide who will get to bat first, or roll the dice hoping your piece won't land where a snake's head is on the board, we have all encountered the mysterious concepts of randomness and probability. But maybe, a coin toss or a roll of a die was too simple to illustrate the weirdness of randomness. So, I will use one of my favourite examples of the weirdness of probability that escape human intuition: the Monty Hall Problem!

1   The Game Show

Let me describe the Monty Hall problem in its original form, a tv show game. It is a simple game of chance, where the player has no control, and wins purely out of luck, or is it?! The game features three doors, and the game show host (let's call him Monty) tells the player (let's call her Alice) that there is a jackpot prize behind one of these doors. The other two doors contain a donkey each that Alice will have to ride and be humiliated on tv. Monty asks Alice to choose one door, and Alice does what she can do best, chooses one door at random.

So far, there is nothing interesting about this game, but the next twist is what defies all human intuition whatsoever. Instead of revealing to Alice what is behind the door that she chose, Monty instead opens one of the leftover doors, and shows that it had a donkey. Monty then gives Alice a choice, "would you like to change the door and bet on the other door?"

And belive it or not, the answer to this question is not trivial! And it is so mind-bending that even though I know the math behind this, it still doesn't make sense to me. Take a moment to answer this yourself. Maybe simulate this as an experiment with your brother using tea-coasters as doors, and paper chits labelled 'prize' and 'humiliation' as proxy rewards.

2   To switch or not to switch

If you said that it doesn't matter what Alice does, since there was a one in three chance that the original door she chose was correct. And even after Monty revealed what was behind a door, the previous fact is still true: there was a one in three chance that the door Alice chose originally was correct. But in fact, out of the two strategies (1) stick to your original choice, or (2) switch to the other door; one is decidedly better than the other.

When I say that strategy A is better than strategy B, I mean that there is a more than 50% chance that by using strategy A Alice will win the prize, and have a less than 50% chance when using strategy B. What that further means is that if Alice played this game many times, say 1000, then she wins more than 500 of those games when employing the better strategy. And since the games that she loses are exactly the games she would have won if she had played using the other strategy, she wins less than 500 games when playing the worse strategy.

Now that I've told you that one of the two strategies is better, I would like you to predict which is the better strategy: to switch or not to switch, that is the question. Even if you can see the correct answer in bold below, I would still like you to think about why it is correct, because who knows, I might be wrong.

The answer is... Alice should switch doors! So, kudos to you if you said switching is better. But why should Alice switch the doors? The math is out there, and can be derived using a simple application of the Bayes theorem. A warning here: as I said, knowing the math does not make it any more intuitive. However, I find that simulating thousands of experiments and seeing our intuitions fail is much more rewarding.

Below is the graph of the simulation of 1000 games, where Alice chooses to switch the door when offered the choice. The height of the bar labelled "Wins" on the x-axis represents the number of the games she wins employing this strategy. Likewise, the height of the bar corresponding to "Losses" on the x-axis represents the games she loses, this also corresponds to the games she would have won by not switching. She clearly wins more than 500 games when switching the door.

3   What does this mean for probability?

There are at least two ways of looking at probability. One of them is that probability tells us about the facts of the world, and what happens when we interact with them in a random way. This view tells us, for example, that if there are 8 red and 2 blue balls in a bag, then the chances of pulling out a blue ball from the bag is low (2 out of 10, precisely). The other view is that probabilities are a way to quantify human knowledge that might be uncertain, thus making our actions based on the beliefs random. This happens when we have partial knowledge of something that is hidden. If someone goes to the doctor complaining of headache, the doctor knows that COVID is spreading and there is a high likelihood that the person has got the COVID too. When the patient says that she was exposed to another COVID positive person, the doctor's belief strengthens and now they believe that the probability that the patient has COVID is higher. The doctor further prescribes a COVID test. If the test results negative, the belief has changed once again, and now the doctor believes it might be just common flu. Note that, these two views are not completely separate, and they are both correct.

The view of probability that we want to prescribe to when thinking about the Monty Hall problem is of the second kind, the one where probability represents our belief of the real world. Alice knows that one of the three doors has a prize. More importantly, Alice also knows that Monty knows which door contains the prize. And this is what Alice relies on. When Monty reveals another door that doesn't contain the prize, he reveals some of the knowledge he had to Alice. This knowledge was previously hidden to Alice, and the act of showing the door is the patient telling that they were in contact with a COVID positive person. It changes our belief. Monty had to just leave one door that didn't have the prize.

How is that significant? Well, the probability that Alice chooses the prize door is still one in three. There is a two in three chance that Alice didn't choose the prize door. The following will be important: this also means that there is a two in three chance that one of the other two doors contains the prize. But once Monty reveals a donkey door, our belief about the world changes. Specifically, the probability that whichever door monty revealed is a donkey door becomes 100%. This leaves one other door that neither Alice, nor Monty chose. And since Monty revealed one of the left-over doors, our belief about the left-over doors is what changes. Specifically, previously there was a two in three chance that original chosen door was wrong (and that one of the two left-over doors contained the prize). Now, there is still a two in three chance that the chosen door is wrong, and that the only left door contains the prize. And we see that Alice wins roughly \(2/3\times 1000\approx 666\) games in the above figure. Choosing a door fixes its probability of being correct, and the opening of a door moves its probability of being correct into the other remaining door because Monty uses his knowledge to skip over the correct door.

4   Monty-Haller than Monty Hall himself

To help you further internalise this scenario, I will take the Monty Hall problem to its extreme. We will increase the number of doors. There will still be only one prize. Of the doors Alice doesn't choose, Monty will reveal all but one and show that they all do not contain the prize. Then, Monty will again offer Alice the choice of switching your door.

As Monty opens each door, their probabilities of being the prize door start moving to the other doors that Alice didn't choose. As Monty reveals all but one door, our belief of the world that the door that we didn't choose is correct strengthens.

I simulated 1000 games with more than 3 doors in the following plot (you might need to zoom a little if you're on your phone), and as you can see, switching the door becomes better and better. This is because Monty gives us more and more information with each door he opens. The more door he opens, the more information we have, and more the probability of prize shifts to the other door.

5   An explanation using the other perspective of probability

The perspective of probability as the quantification of human probability is important, but not very intuitive. Therefore, I present an alternative explanation of why Alice's strategy to switch the door is better in the case when there are three doors.

There is a one in three chance that Alice wins. There is a two in three chance that she loses. Monty always eliminates a non-prize door. In two of the three possible scenarios where Alice was originally losing, she wins by switching the door. And in the one out of the three scenarios, where she was winning, she loses by switching. Therefore, after all this, there is a two in three chance that she wins by switching. She converted her loss probabilities into her win probabilities.

This can now be extended to many doors. For example, with 15 doors, the probability that she wins by switching is the probability that she loses in the original game, which is \(14/15\), and the number of games she should win is around \(14/15\times 1000\approx 933\). Check the graph for how accurate that number is!

Conclusion

I would like to conclude with a word of caution. The caution that this does not apply in a test with multiple choices with one correct answer. If you don't know the correct answer, you cannot choose one of the available options at random, then eliminate all but one, and think that switching is more than a \(50\%\) getting the correct answer. Because in the game we relied on Monty's knowledge of the correct door, but in the exam there is no one to supply this information externally. In conclusion to the conclusion, probability is weird, and we need to be careful about how and where to apply it to win the games we play, be it a tv show, an exam, or life.