## Thursday, June 24, 2010

### Monty Hall problem and its solution/explanation

In this article we’d like to concentrate on a different kind of puzzles : paradoxes! Specifically the Monty Hall problem. It is one that has intrigued mathematicians and common men alike for ages.

Those who have watched the movie 21 might have been baffled by the classroom scene where the professor presents to the students the Monty Hall Problem. It is paradoxical because the answer is not intuitive. The problem statement is thus:

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?" Is it to your advantage to switch your choice?

One important point to note here is that the host knows what is behind the doors. This will ultimately affect the answer. Many people, including some with PhDs, instinctively say that once a door is opened, the probability of the car being behind either door is 1/2. In reality, however, the probability of the car being behind the door you did not initially choose is 2/3. This means that switching your initial choice increases your chances of winning!

Most people go into an argumentative mode when they hear this answer. However, there is a mathematical proof supporting this answer. Since the mathematical proof is too long for this article, we will see the logical proof which is just as convincing.

The simplest way to look at this problem is to divide the three doors into only two parts. Thus, the door you choose is one object and the other two doors group together to form another object. The probability of the door you chose hiding the car is 1/3, while the probability of the car being behind one of the other two doors is a combined 2/3. When the host reveals a goat, he is effectively eliminating one of the doors from the group you did not choose, making the probability of that door hiding the car 0. So now, the entire 2/3 probability of the other group hiding the car is transferred to the unopened door.

When the host offers you a choice to switch doors, what he is effectively saying is “you can keep the one door you have, or you can take both the other doors.” Clearly, taking two doors is always more beneficial than choosing one!

So the next time you are on a game show and the host asks you such a question, you know what to do. And if you never end up in any game show, at least you can baffle your friends with this knowledge of the Monty Hall problem.