The “Monty Hall Problem” is one of the classic mathematical problems which has entertained many and puzzled equivalent number of people. You will agree with me if you are already familiar with it and in case if you are hearing the term ‘Monty Hall Problem’, for the first time, I wouldn’t ask you to shoot yourself. The whole purpose of writing this post is to make more people familiar with this problem. It goes like this…
Imagine you are a participant in a game show. The game is about selecting from three closed doors, behind one is a posh car and behind the other two are goats. Obviously, as a participant, your aim is not to carry home a goat, but the posh car. Upon making your choice, the host, who has prior information about the position of the car, shows you a door, which you haven’t selected and has a goat. Now, he again offers you to make a choice, from the remaining two doors, behind one is a goat and behind other is a car. So, your chances of winning the car shoot up to 50%. As easy as it sounds, right? Now, here kicks in the math which makes this seemingly simple problem complex, easy if you are among those math geeks. Initially, your probability of winning the car was 1/3. That means your probability of losing was 2/3. The moment one of the door behind which there is a goat, is opened, you can increase your chances of winning the car to 2/3, simply by switching your earlier choice. This switch accounts to the variable change, which is like choosing two out of the three doors, initially. Of course a lot of people don’t realize this trick, and fail to take advantage of the second opportunity.
Didn’t I tell you the problem seems easy, but isn’t quite as easy as it seems to be. To make things clear, I created an illustrative picture of the same. So, here let us say, you selected D3. So, the initial probability of winning is 1/3. Once the host shows the goat behind D1, and if you switch to D2 from D1, the combined probability of D1 & D2 still remains 2/3. Finally this switch in choice is what gets you that posh car, behind D2.
Finally, don’t curse me for posting a problem and a solution, none of which I have created. I never claimed that, did I? The only aim to write this article is make more people aware of this classic mathematical problem. Another subtle message which this problem teaches us is, life seldom shows you a small example of the wrong choice that you made. And when it does, switch your choices. It may increase your winning possibility!
PS: If you interested to know more about this problem, please check out the Wikipedia page: Monty Hall Problem
9 comments:
Good post, but too mathematical. You should write on simple topics.
Point taken... Here on 'Say No to Math' will be it. Thx for reading.
Hahaha..... good post ...... but to certain extent went over my head ..... will try to understand once again by referring to the wiki page :P
heyy good one ....!!
Actually I recieved mixed response for this article. Not sure if I should post articles which are in line with this or not.
Nice post. But if the host opens a goat-door after your first pick, aren't your chances 1/2 instead of 2/3?
The assumption is based on the fact that the host ALWAYS opens a goat-door irrespective of if you have picked a car-door or a goat-door on your first guess.
-Tejas
@Tej: The way in which you view this problem is that the initial winning probability is 1/3. This would imply probabilty of losing is 2/3. I mean, regardless of what choice you make, there are two doors that can be combined and one of them will have a goat behind it (based on the Piegon Hole Principle). Now the host will always open that door which you haven't selected and is a goat-door. Now by switching you get to pick a combo of the previously left out doors, which implies a winning probability of 2/3.
This is a very tricky problem and you are right, the assumptions are the key here!! Thanks for your comments.
You have displayed your intellect by writing this article. I really liked it. A good combination of intelligence and writing abilities, you have got sir.
Dear Anonymous,
Your comments are very kind. I've not tried to display my intellect though by posting such an article. In fact there was no planning in posting this one. I'm glad you liked this post.
You can addreess me as Vijay. 'Sir' seems way above an adress.
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