Saturday, May 24, 2008

Monty Hall Problem

To quench the thirst of my interest in probability, I am currently reading Chances are... Adventures in Probability. It talks about lots of probabilistic topics with real life examples and history. On pages 72-73 it discusses the Monty Hall problem, named after the host of the television game show Let's Make a Deal:

"You have been called down from your seat in the studio audience and now stand facing three doors: one conceals a car, the other two a goat each. You choose Door 1, the host now opens Door 3 and reveals a goat. Do you want to stick with your choice or switch to Door 2? Your train of thought would probably go like this: 'There were three doors available; now there are two. I don't know what's behind either door, so it's an even split whether the car is behind Door 1 or Door 2. There's no more reason to stick than to switch'.

Probability takes a different view. When you choose Door 1 there was a 1/3 chance the car was behind it and a 2/3 chance it wasn't. In opening Door 3, our host has not changed the original probabilities; there is still a 1/3 chance of its being behind Door 1 - which means there is a 2/3 chance of its being behing Door 2! You are twice as likely to win the car by switching your choice..."

Update (August 18th, 2011): It is much easier if you think this way (also checkout wikipedia for extensive treatment):
Imagine that there were a million doors. Also, after you have chosen your door; Monty opens all but one of the remaining doors, showing you that they are “losers.” It’s obvious that your first choice is wildly unlikely to have been right. And isn’t it obvious that of the other 999,999 doors that you didn’t choose, the one that he didn’t open is wildly likely to be the one with the prize?
Since I could not believe it, I decided to test this hypothesis by writing a computer program that would simulate the above show. And to my surprise, yes, they are right! In the screen capture below, I did 50 choices, switching my original choice at each run. As you see, my winning ratio was 68%, which is close to 2/3 (67%)


After that I tried the stick-with-your-original-choice strategy and as you can see, my winning ratio declined to 42% which is closer to 1/3 (33%)


If you want to try it out yourself, you can get the executable. You can also take a peek at the source code written with Deplhi 7.

Explanation [The Psychology of the Monty Hall Problem, Figure 1, p.5] (Thanks to Nart):

3 comments:

Nart Bedin Atalay said...

"... not only is it difficult to find the correct solution to the [Monty Hall] problem, but that it is even more difficult to make people
accept this solution" (Kraus & Wang, 2003 p. 4: http://www.usd.edu/~xtwang/Papers/MontyHallPaper.pdf )

Samil Korkmaz said...

A nice remark from Monty Hall paper, p.4:

"...math answers aren't determined by votes"

... said...

Okuduğum bir yazı doktorların olasılığı bilmeden nasıl hastalara karar vermelerini sağlayan açıklamalarda kullandıklarından dem vuruyordu (Meme kanserinde teşhis ve operasyon kararı acayip bir muammaymış okuyunca erkek olduğuma sevinmiştim )) ). O yazıda bu örneği vermişti yazar, zamanında bu sonuç açıklandığında pek Ünvli kadro orada burada yazarak hesabın yanlış olduğunu göstermeye çalışmışlar. Toplum olarak olasılığın karar verme sürecindeki etkisine güzel bir örnekti.

RL