The Monty Hall problem itself is very easily stated: A contestant is faced with a choice of three doors. Behind one door is a car; whilst behind each of the other two doors is a goat. The contestant first chooses one of the three doors. Once the contestant has made a choice, the game show host (who knows what is behind all of the doors in advance) opens one of the remaining two doors to reveal a goat. The contestant then has the opportunity to either stick with his initial choice or to change to the other remaining, unopened door.
Repeated studies have shown that most people decide to stick with their original choice rather than change. It appears that many people feel motivated to remain with their initial “gut choice”. Furthermore, the decision is often buttressed with the (albeit incorrect) assumption that there is an even split in the chances of winning between remaining with the original choice or changing to the other door.
Just like Buridan’s ass?
Many (incorrectly) view the situation at the final stage of the game as being similar to the choice facing Buridan’s ass, which is often used as an illustration in philosophy to highlight an apparent paradox in the conception of free will. Here, Buridan’s ass is placed equidistant from two identical bales of hay; one on its left and one on its right. Since there is nothing apparently to distinguish one bale of hay from the other, the ass becomes fixated, unable to choose between the two identical bales, and finally dies of starvation.
In the case of our game show contestant, however, the agony of being forced to choose between two seemingly indistinguishable choices is alleviated by the comfort, or convenience, of being allowed to stick with the initial decision. Moreover, the trauma that might be experienced in having originally made the correct choice, only to learn later that it was changed at the last moment, is avoided.
Evidence seems to suggest that people (unaware of the best strategy) choose to remain with their initial choice even when given the opportunity to change it. Unfortunately, and perhaps surprisingly, this means that they will just have cut their chances of winning the car by fifty per cent! The chances of winning the car are always increased, doubled in fact, by changing from the initial choice after the game show host has opened one of the remaining two doors.
The situation at the final stage of the game is not the same as that faced by Buridan’s ass.
Information we can use to our advantage is available
Realizing the subtle effect that the availability of information can have on the chances of making the best choice in this situation is the key to understanding the best strategy. This is described in Bayes’ theorem in mathematical probability-theory, which relates current probability to prior probability.
The fact that many, if not most, people, including some with a mathematical background, find this hard to believe, and in some cases vehemently reject it, is quite remarkable. The reason seems to be because they cannot accept that there could be any difference in the chance of winning whether they stick with their original choice or change their mind. In terms of the chances of winning, both choices are often perceived as being equal. Ironically, by sticking with the original choice, the chances of winning are actually much less than even; but by changing, the chances are much greater than even.
A tale of two realities
What escapes the notice of many people is that there are really two distinct realities, or viewpoints, present in this game. A contestant who started the game with the choice of three doors, and who witnessed the game show host open one door to reveal a goat, does not share the same reality as a second, hypothetical contestant who joins the game at the very last stage. This second contestant can be viewed as being only given a choice between two doors, with no other information available, oblivious to what has taken place beforehand. The second contestant is unaware which of the two remaining doors was initially selected by the first one.
The problem is that many people see themselves in the position of the second contestant, and not the first; and this is a mistake. The first contestant has actually more information available about the situation than the second, and can effectively use Bayes’ theorem to increase the chances of winning the car.
The fact that the chances of winning are greater if the contestant always changes his, or her, mind can be explained quite simply. The probability of choosing the correct door at the beginning is 1/3. And, importantly, the chances of choosing the wrong door with the initial selection is 2/3. Both probabilities here must, of course, add up to one since there are only two possible outcomes.
If you choose a particular door and stick with it, this means that the probability of winning, even after being given the opportunity to change your mind, remains fixed at 1/3.
After the game show host has opened one of the doors to reveal a goat, the sum of the probabilities of winning if you either stick with your original choice or you change your mind and chose the remaining door must also add up to one. With this in mind, the probability of winning if you change your mind is hence 2/3. In other words, you have twice as much chance of winning if you change your mind compared to if you stick with your original choice!
The effect of changing your mind at the last stage is even more dramatic in versions of the game with more than three doors. For example, with 100 doors, your chances of winning are 99% if you follow this strategy.
Some similarities with guided-missiles and quantum mechanics
Optimizing your success rate, or improving decision-making in the light of new data or information, is not just limited to strategies for winning game shows. Missile guidance systems, for example, use something called a Kalman filter. Here, the best estimate of the missile’s position (equivalent to making the choice of door with the highest probability of success in the Monty Hall problem) involves making an initial estimate using a computer programming running inside the missile, and then updating the estimate when more information from the missile’s measurement sensors becomes available.
Both the computer prediction and the measurement sensor value have uncertainty associated with them. The Kalman filter combines the initial computer estimate with the extra information from the measurement sensors to produce the best possible estimate, namely the one with the smallest amount of uncertainty associated with it. This is analogous to choosing the door in the Monty Hall problem with the smallest probability of failure, giving you the highest chance of winning the car.
The Monty Hall problem can even be viewed in terms of the weird world of quantum mechanics. Initially, a probabilistic wave function distributes the car evenly behind the three doors (or however many doors are being used in the game). In the case of three doors, the situation can be interpreted so that initially there is 1/3 of car behind each door. In general, as more doors are opened, and more information given, the wave function “collapses” and the car is seen as being more localized. The probability of it being behind a specific door increases. In versions of the game with many doors, this probability increasingly tends towards one.
The Monty Hall problem, for all the simplicity in its description, has been shown to be a rich vein for tapping into some profound concepts in mathematical probability-theory and quantum physics, not to mention also philosophy and psychology.