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Monty Hall & Getting it Right the First Time

Let's talk a little about the Philosophy of Probability, maybe we'll stumble more into Decision Theory, just for giggles[1], specifically, I'd like to think about the Monty Hall problem.

The Monty Hall problem goes essentially like this, suppose you are a contestant on a game show, in particular, one that actually existed called 'Let's Make a Deal', the premise of this game show is that you are to pick a door and you get to pick keep what is behind the door, behind two of which are crazy-eyed goats[2], and behind one is a new car -- one that would be desirable to a person who knew a thing about cars. After you make your first guess Monty opens one of the remaining door and reveals one of the wayward goats, following this you get a choice -- do you stay with the door you initially picked or do you switch to the other door.

If you perhaps suspect that whether or not your switch the door makes no difference you're not alone, but you would, statistically speaking, be wrong. Probability states that you should always switch doors, this is because your initial choice is between two doors, now that one of the suspect goat doors have been removed you have a 2/3 chance of getting the car if you switch, and a 1/3 chance of getting the car if you stick to the same door. You can read a much better explanation of the problem and why the probability is better on the Wikipedia Page. The problem is a very cool one, probably because it seems counter-intuitive.

Of course, even if the probability is greater if you switch doors, it is still not spot on and even if you switch you could be the proud owner of one of the few animals out there that legitimately haunts my dreams. The point of probability is that if you iteratively run the problem over and over again you will begin to see the outcome with the higher probability be determined more often, but not always[3]. This is an approach used in AI, in particular via Bayesian Networks, we use probability and repeated running of a process to find the best solution in a non-deterministic environment. It does require that on top of probability we have time and data to weed out what the best solution is. You can actually use Bayes' Theorem to directly calculate the probability to getting the car.

To me, the interesting part about all this is the natural inclination of people (myself included) to believe that no matter which door you choose the probability is the same that you will be the owner of a new and impressive mode of transport. Something I've talked about a little before is that our ad hoc reasoning skills as they compare to the machine method of perfect information. Certainly if you run the Monty Hall problem 100, 1000, or 10000 times you will close in on the magic 2/3 number. However in the context of the game you only get one shot at the answer, making switching perhaps the best answer but certainly not the perfect one.


  1. I really do think the language in this blog as a whole is going to get more whimsical as time goes on, I can only be rigorous so long before I start to worry I sound pretentious ↩︎

  2. I hate goats, they creep me right out. ↩︎

  3. Obviously ↩︎