The Monty Hall problem
Today, I’ve been watching a movie that a friend recommanded, and in the beginning, they mentioned something, a problem, that I found very interesting and made me a little bit confused : the Monty Hall problem.
The idea is very simple : You (the contestant) are given 3 doors, behind 1 door, there’s a car, and behind the 2 other doors, goats. You are asked to pick one door, and the the host (Monty Hall) opens another door that shows a goat, and ask you, do you want to switch ?
Intuitively, most people would say that there is no difference if you stick to your initial choice, or change it, but mathematics and Bayes have the answer for you :
- The probability that the prize is behind door_1,2 or 3 is :
P(1) = 1/3 , P(2) = 1/3 , P(3) = 1/3
2. Suppose now that you ( or the contestant ) chooses door number 1, what’s the probability of the host opening door number 3 conditional on where the prize is located ? The trick is here, or what they call variable change
P(door_3/1) = 1/2 (The probability of the host opening door_3 given that the prize is behind door_1. Since the contestant has chosen door_1, the host can open either door_2 or door_3 with equal probability)
P(door_3/2) = 1 (The probability of the host opening door_3 given that the prize is behind door_2. Since the contestant has chosen door_1, the host will open door_2, otherwise, if he opens the door_2 the game ends now, and they miss the next break’s ADs revenues)
P(door_3/3) = 0 (The probability of the host opening door_3 given that the prize is behind door_3. Since the contestant has chosen door_1, same logic as before, at least that’s my intuition, that if he opens the door having the car immediately, there would be no point of having a game like this)
3. Now remember, you ( or the contestant ) picked door_1, now, what’s the probability of the prize being behind door_2, given that door_3 was opened?
This is a basic application of Bayes Law:
In our case it will be :
So, the strategy to switch to door_2 is statistically, the best strategy, and revealing what’s behind door_3, is indeed, relevant.
I didn’t stop there, because it could have been something that we cannot generalize to n-doors, and I found this reply that sums up the idea of going beyond 3 doors. But in my modest opinion, this problem has too many hypothesis and assumptions, that it makes it difficult sometimes to consider all of them and the hardest part is not usually doing the mathematics but finding out what assumptions you should make, and that’s why apparently, even big mathematicians ( like Paul Erdős ), remained unconvinced until he was shown a computer simulation demonstrating the predicted results.
I hope you enjoyed it, and I would like to have your thoughts about it in the comments.
PS : This time it was a short one, finally !
PS2 : You can always prefer goats to cars, it is really a personal choice
References
- A youtube video explaining the problem
- The wikipedia page of the problem
- A 1986 MIT study
- An numerical example of the 4-doors version
- And the proof of Monty Hall problem I used
