Monty Hall visualization
posted December 2013
The Monty Hall problem is to me one of the most fascinating probability problem (for it's simpleness and unintuitive results) that got my mind blown since I learned about it in high school.
One day in high school, in my Math class, the teacher told us about that famous problem. Monty Hall was an old and popular TV show in the states were you had to choose a door to open from three different ones. Behind one of them was a car, behind the two others were goats. Obviously, the goal of the game was to win the car (except if you were really into goats, but then I guess you could have bought a lot of those with a car).
Anyway, the tricky part was that when you made a choice, the host asked you to wait before opening it and would open another door, revealing a goat. Then he would give you the opportunity to waive your initial choice and swap door one last time.
Here lies the probability problem. Do you think you would have more chance of winning if you changed your choice?
My math teacher said yes, and I could not believe that, I remember loudly objecting, telling the teacher it was not possible, that it was not logical. I declined what seemed grotesque at the time, I refused to acknowledge such an unintuitive result, such a simple thing, my brain could do the calculation easily so why would you tell me I was wrong on such a trivial thing.
But yes, I was wrong. I knew I was wrong. I was upset at my own mind. I didn't understand how I could be so convinced that changing choice wouldn't change my chances of winning the car. The problem was simple, so simple. And yet my mind couldn't make its way around it.
After many years of training my brain to think differently about probabilities, I can know see how this problem works. 7 years after my first introduction to this problem, I can now grasp a part of it. I understand it, I know the probabilities enrolled in the resolution of this problem, I've learned them at uni and I made the effort to think about that problem quite a lot during those last years. I actually often ask that problem to my friends, to blow their mind. But still, 7 years after being introduced to that problem, I still have troubles finding its probabilities "natural". My brain still cannot process the fact that it HAS to work that way, that the world is turning in that direction and no others.
I hope I didn't send you to sleep with this. If you want to know more about the mathematician who published this result and got insulted by numerous math PHD for being wrong, you can take a stroll on the wikipedia page.
My technique to wire my brain on the right path? Thinking about a hundred doors, 1 car, 99 goats. I open one door, the host closes 98 others. It feels easier to process when told this way, but there is still a part of me, somewhere, that tells me it wouldn't change a thing. Even with 98 doors opened. What is wrong with my brain?
If you still don't believe me, there is a short and visually clear explanation here.
PS: this is one of my go to when I want to be amazed at how unintuitive or how little we know about how things work. If you like that kind of thing, you can also check the twin paradox or the biography of Milton H. Erickson.