NOTE: this post also appears on the wonderful site about crazy coincidence, theoddsmustbecrazy.com.
I have always been amused and intrigued by responses to “The Monty Hall Problem”, especially when I talk about it to audiences with a high concentration of engineers and mathematicians. If you are familiar with it, but you’ve always struggled with an unsettled feeling of “this can’t be right”, read further and let me know if my explanation of the solution helps to alleviate the discomfort. If you are not familiar, I guarantee you will give your brain a workout by reading on.
First posed to statisticians in 1975, “The Monty Hall Problem” is well-known among academics because it still sparks debate. Many seem to think that disagreements about its solution stem from issues in the clarity of the problem, but I contend that it really stems from human flaws in the way that we process information.
I often discuss this problem in statistics and cognitive psychology courses for several reasons. It is a great exercise in probability calculation and it can be used to teach basic mathematical modeling (and its purpose). An added benefit, since almost all of my students were psychology majors, is that it also illustrates a flaw in human cognition as well as a pattern of problem solving. Even a knowledgeable statistician feels the need to run simulations to see the solution in action. Even then, fully grasping the mechanisms behind the answer often requires brute force cognition.
In general, human beings have a very difficult time wrapping their brains around concepts of probability. It is much like a visual illusion; we know that the lines are parallel/the circles are the same size/there is no motion, but we can’t make our brains process it in a way that represents that reality. It’s just not how our visual system works. I hypothesize that one of the reasons that probability is such a difficult field for most people is that it involves theory and models, which are distinct from observations and we must represent them differently in our minds to properly deal with them. Applications of probability often involve switching gears from the realm of models to data or vice versa and this is where I think most mathematicians get side-swiped in The Monty Hall Problem.
In essence, here’s the problem:
You are a contestant on Let’s Make a Deal!and Monty loves your creative costume (a teddy bear carrying a human doll), so he calls on you to make a deal. Monty says, “There are three doors – Door #1, Door #2, and Door #3. Pick one and you get to keep whatever is behind it.”You’ve seen the show (you weren’t just walking down Ventura Boulevard in a teddy bear costume for fun), so you know that it is highly likely that there is a coveted BRAND NEW CAR! behind one of those three doors. If you choose wrong, however, you might end up with an ostrich…
You choose Door #3.
Monty then says, “Let’s see what’s behind Door #1!” and the door opens to reveal one of the many consolation prizes (and product placements), a lifetime supply of Rice a la Roly.
Cool! You might get that car after all!
Well, the show was successful because the shell-game-huckster-style of Monty Hall rarely stopped there. In this case, he does what he often does, offers to let you switch from your first choice (Door #3) to the only remaining option, Door #2.
Should you? Does it matter?
Not the Problem
Before I get into the solution, let me first deflect a common complaint from mathematicians. The most well-known version of the problem, from its Wikipedia entry:
Suppose you’re on a game show, and you’re given the choice of three doors: Behind one door is a car; behind the others, goats. You pick a door, say No. 1 [but the door is not opened], and the host, who knows what’s behind the doors, opens another door, say No. 3, which has a goat. He then says to you, “Do you want to pick door No. 2?” Is it to your advantage to switch your choice?
This version does not specifically state the name of the show or indicate the way that game shows of its era worked. If you have never seen the television show (i.e., you are younger than 35), or any game show of its kind, let me explain. Monty is in control of almost everything that happens. The only thing “contestants” can do is make choices when Monty offers them. As you will see, they had more control over their odds of winning than once thought, but Monty manipulates some of the build-up by choosing which items to reveal at different steps in the game.
Unfortunately, many probability theorists and mathematicians took issue with the lack of clarity in the problem (context is important sometimes). This provides a face-saving ‘other version’ for the geeks who get it wrong the first time. But whenever I hear comments like, “Okay, given this version, that Monty knows where the car is.” I usually think, “Of COURSE he knows where the car is! There is no other way to play the game!” and wish that people were more able to accept that they are just as human as everyone else.
The problem itself is written clearly, though: it specifically states that a door without a car behind it is revealed before you are given the option to switch. If the situation was a fully-randomized, double-blind game (like “Deal or No Deal”), then the option to switch would not even be on the table if the car is behind the revealed door. There would be no problem in that case. Therefore, the problem is a question of whether you should switch in a controlled setting – one in which the only participant who doesn’t know the location of the big prize is you.
The issue of knowledge is a factor in our processing of the problem, but it’s not what Monty knows that matters. It’s what you (the subject of the problem) know.
So, let’s put that complaint behind us and get back to the problem.
The Answer, and How to See it for Yourself
Hopefully, if the problem is new to you, you’ve spent some time trying to solve it instead of going with your first gut feeling, which was probably, “It doesn’t matter.”
It does. You should switch.
If you don’t believe me, try running some simulations. You’ll have to run a lot in order to get a large enough sample to be certain to see the trend, but here are a few ways to do it:
- Use your favorite program (MATLAB, R, etc.). There is a good database of pre-written simulators for this here. I am partial to Excel myself, even though it’s a bit more cumbersome. I just don’t remember enough code to use another program.
- Use a web-based simulator. Do it at least a hundred times, choosing to switch for half of the trials, and keep a tally your results.
- Use a die to simulate the outcome, assigning 1-3 to “Door #2″ and 4-6 to “Door #3″ (e.g., if you roll a 5, Door #3 is the one with the car). Roll at least a hundred times, choosing to switch for half of the trials (before rolling!). Keep a tally of the results.
What you will see is that switching will result in winning a car in approximately 2/3rd of the trials while staying will only provide a win in 1/3rd of them.
I know what you’re thinking. “But, there are only two doors left, so it should be 50/50!”
Why it is so Difficult to Accept
Human cognitive development is an interesting process. We learn to interpret information from the environment very quickly so that we can respond to that environment, but learning to reason hypothetically takes more time. Even adults with scientific training have a difficult time separating the concept of variables (each has a set of possible values) and data (values which are known).
In practice, hypothetical situations are often conditional (e.g., “If A, then B”). We tend to use information about what is to reason about what could be. We do this because it often works, but it is one of the ways in which our brains can lead us astray. For example, given the premise, “If I study, I will get a good grade on the exam”, what is the most sound conclusion when presented with a good grade? The most common response is, “I must have studied”, but that is not sound. In this premise, studying provides a guarantee for a good grade, but there is no statement that studying is the only way to get a good grade. It does not, for example, read, “If and only if I study…”
In the case of the Monty Hall Problem, the probability of winning is set before you pick a door. No matter which door you choose, the probability is 1/3rd. This is because there is a 1/3rd probability that the car is behind the door you chose given the information you had when you chose it. In reality, the car is behind one of the doors, so the probability it is behind Door #2 is 100% if it is there and 0% if it is not there. Probability is not a useful way to discuss what is or what happened; it is a tool for predicting what is likely to be true/happen.
The new information provided by revealing a loser changes the circumstances and this where we get trapped in our representations of models and data, possibilities and facts.
You had a 1/3rd chance of winning because there were three, equally-likely locations to choose from. It seems as if cutting the choice down to two should change the odds of winning to 1/2. It seems that way because we are focused on the probability that a given piece of information is true (e.g., that the car is behind Door #1) and not the probability that an event will occur (e.g., that we will win the car). The probability that we will win the car relies on the number of possible states of reality. This, in turn, initially relies on the number of locations for the car. When the situation changes, we try to adjust probabilities based on possible locations (which has changed) rather than on the number of possible states of reality (which has not).
Basically, when Monty makes the second offer, the offer is to switch from the door we have (#3) to the door we don’t have (#1 or #2). It does not matter that only one of those doors is left; there is still only a 1/3rd chance that our door has the car and a 2/3rd chance that the set of the other two contains the car.
If you change the way you represent the problem from the beginning, the solution might seem more reasonable. Specifically, instead of thinking in terms of assigning probabilities to doors, think in terms of assigning probabilities to outcomes: winning verses losing.
Let’s go step by step…
Monty asks you to pick a door from three choices. Behind one of those doors is a car. There are three possible locations and it must be in one of them, so there are three possible states of reality.
You choose to bet on Door #3; there is a 1/3rd chance that you will win the car.
There is a 2/3rd chance that you will not win the car.
This would be true no matter which door you chose.
Monty reveals that one of the remaining doors is a loser. At least one will be a loser since there is only one winner and you can choose only one. The car, however, does not move. Even though there are only two locations left, so there are still three possible states of reality. What’s changed is that we now know more about each of those possible states (there are fewer locations for the car to be):
So, if we model the problem in terms of the probability of winning with Door #3, the model itself does not change after the losing door is revealed. What changes is that we would no longer want to choose that door, so it is no longer among our options. This leaves us with only two options: keep the door we have or switch to the remaining door. The odds of winning/losing with Door #3 have not changed, but eliminating an option allows us to make a better choice – switch.