Saturday, September 3, 2011

The weird problem of MONTY HALL

Today I am going to tell you a story.... A story that I felt was weird ... It is about about the MONTY HALL. Here it goes .....





Here is the Problem. I am in game show. The rule of the game show goes something like this. There are three doors. Behind one of the doors is a car. Behind the other two doors is a goat. So totally there are 2 goats and 1 car. Whichever door I choose, I win the thing behind it. I would certainly want to win a car as I am not interested in the goat. Even if I do win a goat, I win something and I am not completely unhappy.

So I am asked to pick a door. I pick door '1.'. Ok ... My chance of winning is 1/3. 33.33% of winning a car. I keep calm. 

Now there is a twist in the game. The game hose declares another rule. He decides to give me another chance but, this time with a twist. Door numbers 2 and 3 are unpicked. He now offers to open one of the doors 2 or 3 behind which is a goat. He opens door 2 and there is a goat. The game host now gives me another chance. I can either change my door selection or I can go with 1. I calculate that the probability of me winning the car is 1/2 or 50%. A much better probability, but it doesn't really depend if I actually change my option. I still will have 50% chance if I choose either of the doors. I stick with door no.1. The game host asks me if I am sure and I say I am. The game hose opens the door and there is a GOAT !!!!! I had lost the game .... Behind the door 3 was the car !!!!!

What went wrong !!!! May be analysis was wrong or my luck was bad !! The guy after me also lost. So did the guy after him. The guy after him won ... There was nothing wrong with the game I guess ... My luck and analysis was bad was what I thought ....

More careful analysis reveals what went wrong and why !!! The explanation is very simple and should be viewed as a whole and not individually. Then it becomes clear as to where I went wrong ......



The first mistake : What I saw was that the probability of selecting a car was 1/3rd and what I did not see was that, inspite of 3 doors, the probability of me selecting a goat was much higher. it was 2/3rd or 66.66%. So, it now became clear to me that 2 out of 3 times I would have ended up selecting a goat and not swapping next time would clearly be a big mistake !!


The table below shows as to why changing the selection is a good idea. Interesting the chances of winning goes up from 33.33% to 66.66% if you swap. (Refer the figure below as to why)





Now that this problem proved that swapping is actually a good idea, I started thinking about the personality of the people in the game and their winnings. A person who seems to be a stubborn kind (one who sticks with his choice) will end up losing while a person who seems unsure and doesn't really know what to do and changes his option will end up winning 66.66% of the times.