A simple explanation of the Monty Hall problem

You are participating in a game show. There are 3 doors in front of you; 2 are empty and 1 contains a prize. You are asked to pick a door. You do so, but you don’t open it yet. The game show host—let’s call him Monty Hall—now opens another door; not the one you picked. You see that it is empty. You are asked if you want to stick with your choice or switch to the remaining closed door? What should you do if you want to win the prize?

This is the famous Monty Hall problem. I first heard of this problem in high school while reading The Curious Case of the Dog in the Night-Time. I don’t remember any of the rest of the book. But I remember this problem and talking about it with my friends, my parents, my sister and constantly infuriating everyone.

It seems like you have to pick between a door that has a prize and one that is empty. So the probability should be 1/2 of winning. So it seems like it shouldn’t matter if you stick to your choice or switch.

But another argument goes like this: when you first picked a door, it was more likely that you picked an empty door. If you picked an empty door, then Monty has to open the other remaining empty door. Thus, the door that he did not open contains the prize. Since with probability 2/3 you picked an empty door at the beginning, if you switch, then you will win with probability 2/3.

What’s going on?

The key to resolving the confusion behind the problem, is to realize that the best strategy depends on what knowledge Monty had when he picked his door. Let’s consider two possible scenarios:

(1) Monty knows which door contains the prize, and therefore opens only a door that doesn’t contain the prize.

(2) Monty does not know which door contains the prize and he opens one of the remaining doors and at random. Even though you see that it is empty, Monty could’ve opened the door with the prize.

These two cases have different answers! In Case 1, you will win with probability 2/3 if you switch. While in Case 2, you will win with probability 1/2 if you switch. Let’s see how.

Case 1. Monty Hall knows where the prize is and opens the door that’s empty:

Please see the figure below.

Let us label the doors A,B and C. Let door A contain the prize: this assumption doesn’t matter, because what we call A, B or C doesn’t matter. I indicate the door containing the prize with P.

Consider many, many copies of your universe, and you are playing this game in all of these different universes.

In 1/3rd of the universes—Universe 1 in the picture–you pick door A; in 1/3rd of the universes—Universe 2—you pick door B; and in the remaining 1/3rd —Universe 3—you pick door C. In the picture, the door that you pick originally is indicated in green. (Note that each Universe 1,2 or 3 contains many sub-universes).

Thus, in 2/3rds of the universes—Universes 2&3—you have picked the door without the prize. In these Universes, Monty has to open the remaining empty door. We indicate the door Monty opens by blue.

Thus, in Universes 2&3, Monty, by making his choice, has given away information about where the prize is. Only in Universe 1 is he free to open either of the two doors, and he doesn’t give away any information.

The case where the host knows which door contains the prize. P indicates the prize. Green indicates the door you originally picked. Blue indicates the door picked by the host. A,B,C are the door labels.

The case where Monty knows which door contains the prize. P indicates the prize. Green indicates the door you originally picked. Blue indicates the door picked by Monty. A,B,C are the door labels.

It is clear from the figure that you should switch doors in Universes 2 & 3. Only in Universe 1 should you stick to your original choice.

Thus, in 2/3rds of the Universe if you switch, you will win the prize.

And since you should always behave as though you are in the most likely universe, you should switch. And you will win with probability 2/3.

Case 2. Monty Hall does not know which door contains the prize:

Again, please see the figure below.

In 2/3rds of the universes, you have picked the empty door (indicated by green). In the remaining 1/3rd you picked the door with the prize.

In the universes where you picked the door with the prize, Monty will always open an empty door (the door he picks is indicated in blue). So far, so good. This is Universe 1 in the picture.

But, in the universes where you picked an empty door, among the doors that remain there is one empty door and one door with the prize. These are Universes 2&3 in the picture.

Now, because Monty does not know which door contains the prize, in half of the universes he will open the door with the prize. This is Universe 3 in the picture. And in the other half he will open the empty door, which is Universe 2.

But we know that we are not in Universe 3 because we see that Monty opens an empty door.

Thus, we must be either in Universe 1 or 2. It is clear that the if we are in Universe 2, we should switch and if we are in Universe 1, we should stick to our choice.

In this case the host does not know where the prize is.  Again, green indicates the door you picked, and blue the door picked by the host. And P indicates the door with the prize. Notice that in Universe 3, the host opens the door with the prize.

In this case Monty does not know where the prize is. Again, green indicates the door you picked. Blue indicates the door picked by Monty. P indicates the door with the prize. Notice that in Universe 3, Monty opens the door with the prize, therefore we can’t be in Universe 3.

But the number of sub-universes in both Universe 1 & 2 are the same!

Thus, we are equally likely to win the prize irrespective of whether we stick with our original or switch! Therefore in this case, the probability of winning is 1/2 either way.

The Monty Hall problem beautifully illustrates how the processes behind the evidence that we see is crucial in deciding how we go forward.

Acknowledgements: Kenny Easwaran pointed out the difference between the two cases when answering a question I asked at a physics department colloquium. I think that’s when the Monty Hall really clicked for me. Also, reading Eliezer Yudkowsky  really clarified some notions of probability relevant to this problem for me.


Why is there no physics forum at the level of Mathoverflow?

If you aren’t already aware, some of the highest-level discussion about math can be found over at Mathoverflow. The questions are all research-level. The quality of answers is very high and some of the best mathematicians in the game, including several Fields medallists, routinely participate in the discussion.

Here is a puzzle then: why isn’t there any physics forum operating at this level? Physics StackExchange is good, but nowhere close to Mathoverflow.

I offer a few, not-mutually-exclusive, hypotheses:

1. People don’t understand physics as well as they understand math. Especially everyday physics, such as: why are clouds white or why do all the planets orbit in the same plane. These require a combination of physical intuition and mathematical ability. In this sense, physics is harder than math: it is easier to frame interesting physics questions which have difficult answers than it is to frame interesting math questions with hard answers (Yes, number theory is an exception). Indeed, you need little physics training to ask why the sky is blue or why do sand-dunes form.

2. The language of math is easier to communicate in. You need less words and more symbols in math. But in physics, you need more words. So you need people who are especially clear in communicating physics ideas. This is a skill that is harder to acquire than communicating mathematical ideas. Therefore, if you compare a physicist and a mathematician who both understand their domain equally well, then it is more likely that the mathematician communicates better.

3. Curiosity about math is easier to develop than curiosity about physics. I know this sounds counter-intuitive. But I’m talking about deep curiosity, the kind of curiosity that makes you explore answers yourself. Deep curiosity about physics requires a kind of naive curiosity about everyday things: you need to look around, go pick up things and play with them. This is socially visible and this kind of naivety is looked down upon; it is the opposite of nil admirari. But deep curiosity about math is easier to develop. You can do it in privacy. Just pen and paper. Thus, less people have developed a good intuition about physics.

4. Curiosity about math is more easily rewarded. The answers you get are clear and satisfying. But in physics, you are never sure if the explanation that you came up with corresponds to reality or whether you’re missing some important subtlety. To check, you have to find a way to test your hypotheses with real world observations. In the case of math you can get a crisp answer: a proof or a counter-example.

5. The physics education system isn’t very good at inculcating all of these abilities because they are harder to teach. Interestingly, on internet physics forums such as the old Orkut physics forum and Physics Stack Exchange (I’m not sure about physicsforums.com) the best explainers were, in some sense, outsiders (i.e. outside academia). Ron Maimon comes to mind. I forget the name of the best explainer in Orkut physics, but I remember clearly that he was a high-school dropout.

6. Historical reasons. Mathoverflow just became more famous because of all the famous mathematicians who came there. But, many famous physicists did try to come to Physics StackExchange as well.  Examples: t’Hooft, Shor, Preskill, Gottesman.

This smells of opportunity. If these abilities: naive curiosity, thinking about more unstructured real life problems using hypothesis testing, intuition, and a mix of math and numerical estimation,and the ability to communicate clearly with both words and equations, are rare, then that means that this is a potentially a rare and valuable skill worth developing.


Book Review: ‘Violence’ by Randall Collins

Randall Collins is an ambitious sociologist. His aim is to build a comprehensive theory of violence, in all of its different manifestations. This includes violence in military contexts, police brutality, mugging, bullying, domestic abuse, violent carousing, violent sports such as boxing, violence during sporting events such as fights during a baseball match or audience violence after a football match, dueling in the 19th century, and even mosh pits.

The thesis that connects all these manifestations of violence? Violence is very hard for humans.

He proposes that all human beings whenever put in a situation that is potentially violent, come up against a wall of “confrontational tension and fear”. This is his primary theoretical construct. The source of this confrontational tension/fear is not explored in detail; he proposes that it is a consequence of attempting to override fundamental human instincts towards mutual emotional entrainment and instincts towards engaging in solidarity rituals. Importantly, it is not just fear of injury or death. This is evidenced, for example, by the fact that soldiers in battle experience much more fear than medics in battle, though they have similar exposure to danger.

If we accept this fundamental difficulty to committing violence, then his task is to illustrate the situations in which some people are able to overcome the tension/fear and proceed to violence. His focus is always on the situation and far less on background factors such as race, socioeconomic status or criminal history. As he repeatedly points out, background factors only account for very little in the causes of violence: most poor people do not commit crime; most criminals are not violent; most drunken people do not carouse violently; most police arrests do not turn violent; most young men are not violent; most child-abuse victims are not violent and so on.

He proposes different and varied situational pathways that allow people to overcome confrontational-tension/fear. For example, most police brutality incidents, such as the famous Rodney King incident, can be seen as a case of ‘forward panic’: a situation where tension builds up—the high-speed chase, in the case of Rodney King—due to the threat of violence and it is released all of a sudden when one party—the police, in this case—realize that they are much more stronger than the other party—King, in this case. The released tension leads to ugly and brutal violence unleashed by a strong party upon a weaker party.

Forward panics produce the most viscerally ugly forms of violence: take the classic example of police beating up a lone protestor. The Rape of Nanking is another famous example, and is analyzed in the book. The Jallianwallah Bagh massacre also comes to mind, though it is not mentioned in the book.

In bullying and in domestic abuse, the confrontational-tension/fear is overcome by repeated emotional entrainment. The bullied—over a period of time—get trained in their relation to the bully. They ‘learn’ to play the role of the victim. Collins points out that most bullying happens in “total institutions”: closed-off institutions whose status hierarchies do not change over time, and there is little opportunity for participants of the institution to go somewhere else. The classic examples: prisons, high-schools and families. In total institutions there are more opportunities for both bullies and the bullied for repeated interaction and thus repeated emotional training.

And similarly, he dissects dozens of forms of violence. The overarching theme is that violence is hard. Violence needs certain situational variables to be conducive. And even when it is conducive, violence is usually limited to a very small number of people and is generally incompetent. A striking example: on average, only 15% of frontline US Army troops during World War II even fired their guns.

As any good theorist, he realizes that there are exceptions to any rule and he tries to understand them. For example, some military snipers have a fantastic record of kills, far more than most people in the fighting force. Similarly, ace pilots and famous mafia hitmen. All of these are among the very few people in the world who are competently violent.

What situational dynamics makes this possible? Snipers for instance, operate under cover, very far away from the enemy and never making eye contact: this allows them to overcome confrontational-tension/fear. Similar mechanisms are proposed for other competently violent people.

Overall, this is a fantastic book. It is beautifully written and the language is kept as plain as possible. I’m not a sociologist, but I was able to understand most of this book clearly. Whether his theory is successful or not is an open question.

His analysis is always honest, and he is always willing to look at the exact places where his theory seems to fail. And he is willing to accept the parts that his theory does not explain. In fact, he hints at a much broader theory that would simultaneously account for both background and situational variables.

I worry whether some of his explanations about how confrontational-tension/fear is overcome aren’t too contrived. He repeatedly points out that most situations that have conflict do not proceed to violence. And he attempts to clarify the situational dynamics which allows violence. While definitely he goes some way in explaining the dynamics, I don’t know if he really completes the picture. But then again, he says that he is setting up a companion volume to this.