Suppose you had the opportunity to be on the famous 1970s game show “Let’s Make a Deal”, which was hosted by the great and often witty Monty Hall. You have the option of choosing between three doors, behind one of which is a new car. Select door number 1. The resourceful Monty Hall decides to open door number 2 to reveal a goat. Now gives you the opportunity to change your door selection. The question is, do you change or stay with your original selection? Well, a knowledge of mathematics, and in particular variable change, tells you what to do.
For those of you reading and processing this, you may have told yourself that you saw this problem appear in the recent release of the movie “21” starring Kevin Spacey and Jim Sturgess. Actually, this is the famous Monty Hall problem that is presented to Ben, the character played by Jim Sturgess, by Professor Rosa, played by Kevin Spacey. Ben correctly solves the problem in the movie and, as a result of impressing his math teacher, is ordered to join the secret MIT blackjack team. Now if you saw the movie, you know what Ben did: he switched doors (oh, and he joined the blackjack team). But why?
Let’s take a closer look at this problem and the reason will come to light. By the way, this is not obvious, and after reading this, you may still be scratching your head as to why the change should be made. However, if you think carefully and reason about it, the motivation behind the change will eventually become clear. So let’s start reflecting.
Behind exactly one door is a new car. There are three doors. Behind the other two doors are hungry but benign goats. The probability of selecting the door with the car is obviously 1/3, and the probability of selecting a goat is then 2/3. Now I stated that the clever Ben Campbell from the movie was right to switch, and I explain to his teachers and classmates that by virtue of the change, he increased his chance of landing the car to 2/3 from 1/3, courtesy of the benevolence of his teacher. But why?
The easiest way to look at this, I think, is this: the probability of selecting the correct gate is 1/3. The probability of losing is 2/3. You have selected door number 1, which has a 1/3 chance of having the car. You have a 2/3 chance that you don’t have the car. When Monty Hall reveals the goat behind one of the other doors, he has essentially transferred 1/3 of the probability to him if he makes the switch, now giving him a 2/3 chance of getting the car.
As you say? Okay, separate the test on the door you select and the other two doors. The probability that the car is in one of those two doors is 2/3. It has been shown that a door does not have the car. When switching, you now have a 2/3 chance of getting the car because you now have one of the doors in the group with a 2/3 chance. That’s all about it.
Yes, I know. You may be scratching your head thinking, “I don’t get it.” Go back to the last paragraph and reread it several times. Think about it. The answer will surely come. Math: Yes, that’s the real deal.