The Monty Hall Problem
by Carlos Andre Reis Pinheiro, Data Scientist, Teradata
The Monty Hall Problem is a very good example of how important it is to be well aware of activities in the marketplace, the corporate environment, and other factors that can influence consumer behavior. And equally important, it illustrates how critical it is to understand the modeling scenario in order to predict activities and events. In order to increase your chances, particularly from a corporate perspective, it is important to understand the equations that, at least with reasonable accuracy, explain the scenario under investigation—even if that scenario is largely composed of a situation that is dependent on chance. It is then possible to at least create an expected outcome of a particular scenario, whether it be probabilistic (based on historical information of past events) or stochastic (as would be with a sequence of random activities). Breaking this down, if you are going to flip a coin, and you are about to bet based on it, you should know your chances of winning the toss are about 50-50, no matter whether you choose heads or tails. Although this sounds quite simple and straightforward, companies don’t do this very often. Companies typically do not prepare themselves for upcoming events—gambling even more than they should. The Monty Hall Problem is a case that illustrates this notion, that the knowledge about the scenario and the chances involved make all the difference between winning and losing.
Let’s Make a Deal is a television game show originally produced in the United States and thereafter popularized throughout the world. The show’s premise is to have members of the audience accept or reject deals offered to them by the host of the show. The members of the audience who participate in the game usually have to analyze and weigh all the possibilities assigned to a particular offer described by the host. This offer could be a valuable prize or a totally useless item. Monty Hall was the famous actor and producer who served as the host for this game show for many years.
The Monty Hall Problem is in fact a probability puzzle, considered to be a paradox because although the result seems impossible it is statistically observed to be true. This problem was first proposed by Steve Selvin, in a letter to the American Statistician in 1975. This problem was published again in Parade Magazine in September 9th, 1990, within the Sunday “Ask Marilyn” column, on page 16.
“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, 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?” The question is: Is it to your advantage to switch your choice?
In her response, Marilyn vos Savant said that the contestant should always switch to the other door and by doing so, she explained, that the contestant was going to double his or her chances. She was flooded with more than ten thousand letters from all over the country, including letters from five hundred PhDs, mathematicians, and statisticians associated with revered and prestigious universities and institutions. The majority of the letters expressed concerned with Savant’s mistake, and some of them even asked her to confess her error in the name of math. Intuitively, the game concept leads us to believe that each door has a one-in-three chance to have the car behind of it, and the fact that one of the doors has been opened does change the probability of the other two. Once one door is opened, the chances of having a car behind one of the other two doors are one in two for each one left. When one door is opened it is like we have essentially changed the original scenario, from three doors to just two. But in reality, there is no changing the probabilities for each door, either before or after the first door has been opened. All doors keep the same one-in-three chance of having a car behind of it, both in the occasion of the first question (three doors closed) and on the occasion of the second one (two doors closed and on opened).
Marilyn vos Savant stated that if the car is equally likely to be behind each door, a player who picked door number one and doesn’t switch, has a one-in-three chance of winning the car, while a player who picks door number one and does switch has a two-out of-three chance. In this way, the players must switch the door to double their chances. It is quite simple, isn’t it?
The Monty Hall Problem demands some assumptions. The first one is about fairness. The car should have an equally likely possibility of being behind any of those three doors, and the contestant can initially choose any door. Monty will always open another door, giving the contestant the opportunity to switch from their initial door choice. And finally, and mostly important, Monty knows what is behind the doors—and will always open a door that reveals a goat. If Monty revealed the car the game would otherwise be over. By revealing the goat behind the door, Monty doesn’t alter the chances of the remaining doors to 50-50. Actually, and from the beginning, each one of the other two doors not chosen has a one-in-three chance and, therefore, the sum of the remaining doors is two out of three. The fact that Monty opens a door he knows has a goat behind it puts all two thirds of the chances to win in the door not yet picked by the contestant.
You can also understand this problem by calculating the probability of losing. Switching the door makes you lose only if you have previously picked the door with the car behind. And initially, you have one in three chances to pick the car. Hence, if you switch the door, you have a two in three chance to not lose (1/3 for each remaining door).
To finish this up, let’s think about the details of this problem illustrated in a table showing all possible arrangements of one car and two goats behind a total of three doors, as shown in the following Table. Suppose you pick door no. 1. In the first row of the table, you can see that if you switch you lose and if you stay you win. In the second row, if you switch you win and if you stay with your original choice, you lose. Finally, in the third arrangement, if you switch you win and if you stay you lose. By choosing door no. 1, if you keep the door and stay with that choice, you win one time. But if you switch you have a chance to win two times. Analogous calculus can be made for doors 2 and 3. At the end, by switching the door you should double your chances to win. The following Table shows the options and the chances in keeping or switching the doors.
Itemized Choices in the Monty Hall Problem
|Door 1||Door 2||Door 3||Switch||Stay|
The point is that in spite of this particular problem being about gambling, by knowing the proper theory and by understanding the correct way to calculate and model in the scenario, you can double your chances of winning. The same thing happens with analytics.
In spite of the heuristic influence over real-world events, by knowing the proper way to model the business scenario you can substantially increase the chances of applied model success.
In theory, if a company offers a particular product to a customer, the chances that he or she will get that product should be 50-50. But if the company knows about this particular customer, their past behavior, the current behavior for similar profiles, products, and services held by similar customers, and so on, then the company could substantially increase the chance of succeeding in this selling process.