The Monty Hall puzzle and statistics
Have you ever watched the popular American game show “Let’s make a deal”? This show premiered in the US in 1963 and is still running today. When Monty Hall was the show´s host, a successful player would get the chance to face 3 closed doors and choose one of the doors, at the end of each day’s show. Monty Hall would explain to the player that there was a desirable prize behind one of the doors – perhaps a new car or half million in cash – and a goat behind the other two doors.
If the show had just ended here, it would not have been exciting enough for the player, the audience and the statisticians!!! Since, in this case, everyone knew that the probability of the player winning the prize is exactly 1/3. And it is all about luck. If the player is lucky, he/she would win, and would get nothing otherwise.
So, of course the game did not just end here. It had a twist! This twist has delighted statisticians ever since (But perhaps not the players!). After the player chose a door, Monty Hall would open one of the remaining two doors which always revealed a goat. For example, if the player chose Door 1, then Monty would open Door 3 and show that there was a goat behind Door 3, but Door 1 and 2 remained closed. Monty would then asked the player whether he/she would like to change his/her mind and switch door (In this case, switch from Door 1 to Door 2). The question is: should the player switch?
It was not hard to imagine that the audience sitting in front of TV would shout “NO! NO! NO! Do not switch! It is just a trick by the host!” On the other hand, a group of statisticians would shout “Switch, definitely! You would have a higher probability to win the prize if you switched!”
Who is right and who is wrong?
Our intuition probably tells us that “not switching” makes more sense. However, the truth is the opposite! The statisticians are actually right about this! But why?
Let us consider the first case; if the player does not switch, then the probability of winning the prize is 1/3. What happens to the winning probability if the player does switch? It increases to 2/3! But again, why? It is simple. Initially, you have three options (3 doors), and Monty Hall has shown you that Door 3 has no prize. Therefore, you have 2/3 probability to win the prize! This is clearly demonstrated in the graph below. There are only three possibilities: (1) The car is behind Door 1 (2) The car is behind Door 2 and (3) The car is behind Door 3. Let us assume that the player chose Door 1:
From the graph, it is obvious to see that “switching” is always better than “not switching” after a door without the car behind it is opened, since if the player does not switch, the only chance he/she has of winning is when the player initially chose the right door with the car behind it. However, if the player switches, he/she would only lose if the initial choice is right, otherwise, the player would win after making the switch.
So the lesson learnt here is that once you find yourself in this type of the game, make sure you switch! And more broadly speaking, do not let your gut instinct lead you to make the wrong move!
Author: Ming-JinJiang
