Friday, November 03, 2006

The Monty Hall Problem

Imagine yourself on a game show. You are the contestant and you have to pick a door out of the three doors in front of you. The host says that there is a prize behind only one of these doors. Now, you choose a door. The host then opens a door which is not the door you've chosen. Voila!! there is no prize behind that door. After all this drama, the host gives you a choice. you can keep your door, or you can change your choice to the other unopenned door. Faced with this seemingly ambiguous situation, what will you do? Which choice will give you a better chance of winning?

Now, for most people, the intiutive answer will be that it doesn't matter, because there are two doors and there is an equal probability that the prize is behind one particular or the other. So, the answer is 50% for each case.

As you may have guessed that this answer is wrong. And here is the explaination:
Suppose, instead of 3 doors there are a 100 doors. You select one of these 100 doors. The probability that you chose the 'prize' door is 1%. And the probability that you didn't choose the prize door is 99%. Out of the 99 unchosen doors, the host opens 98 doors, non of them containing the prize. With this fact in mind, we can speculate what is happening. The probability of the prize being contained in one of the 99 doors was 99%, but now we know that 98 of those doors do not contain the prize, but this does not change the fact that your chosen door's probability of being the 'prize' door is 1%. What has changed is the probability of the lone remaining door among the unchosen doors. Now its probability is 99% of being the 'prize' door. So, clearly, you'll be much better off changing your choice rather than staying with your current choice.
And what is true for 100 doors is true for 3 doors as well. Here, the probability of winning is 66.66% if you change your choice.

Some things are not what they appear at first.

4 comments:

Anonymous said...

lots of doors .. with an opening of a lifetime at the very end ... :)

keep up the gud work ...

my zodiac kin ...

Siddharth said...

interesting stuff.. nice blog

Abhishek Mishra said...

keep up the good work bro,
ideamonk.blogspot.com

Siddarth Malhotra said...

I wonder how no1 used smthin inside their cranium skulls to give it a second thought. See m no expert but i feel that its fraud ( the % probability :P ) Wanna know why, read on :-


in the end when u mentioned that the probability of it bein in the one of the 99 non chosen doors, after havin opened 98 doors is 99%, well that's assuming that it was there in one of those 99 doors( if that is the case it would be 100% now ), coz if it wasnt, and in the end u have just 2 doors left, u r brought back to it being a 50-50 chance.

U cant calculate it's probability in the start n then keep on using it till the end, the probability changes with every non chosen door being opened ( it's dynamic in nature, in this situation ). So what u mentioned in the entire post is just fake, unless u can prove me wrong :D


Oh btw, i think tu pehchaan toh gaya hi hoga ki main kaun hu ;)