The Monty Hall problem is an interesting illustration that people cannot understand the Bayesian probability rule. This puzzle originates from a TV game show, hosted by Monty Hall which started airing in 1963. At the end of the game the contestants were put in front of three doors. Their presents were behind these doors. Behind one of the doors was a luxury car and behind the two other doors there were goats. The contestants had to choose one door. After he closed the door, the host Monty Hall, opened one of the two doors which the contestant did not choose and it contained a goat. Now, Monty Hall offered a deal to the contestant, would you like to switch the doors? You can give away the door which you have chosen and pick the other door which is still closed. During the show, most of the contestants did not accept the deal, they stuck with their first choice. Out of 228 contestants only 13 percent chose to trade the doors.
But was it the right decision? No it was not! In fact, cognitive psychologist Massimo Piattelli Palmarini has written in his book “The Power of Logical Thinking” that: “No other statistical puzzle comes so close to fooling all the people all the time [and] even Nobel physicists systematically give the wrong answer, and that they insist on it, and they are ready to berate in print those who propose the right answer”.
Why was it in the best interest of the contestants to switch the doors? There is one third probability that the luxury gift car is behind each door. When a contestant chose one of the doors. There is a 33.3333 percent probability that she wins the car, not the goat. While there is 66.666 percent probability that the car is behind one of the doors which she did not choose. When Monty Hall opened one of those two doors, which contained a goat. There is 66.666 percent probability that the other door out of those two contains the car. Therefore, if the contestant stuck with his initial choice, the probability of winning was one third. However, if she switched the doors, her chance of winning the car would have been two thirds.
Yet most people do not acknowledge that the chance of winning for the door which they did not choose in the first place is much more.
Psychologists attribute this to the “equality assumption”. People tend to believe that probability is evenly distributed among all unknown choices. However in some situations, specifically in the choice of Monty Hall show it did not.
