Cap takes on the Monty Hall Problem

17 May 2010

I’m teaching probability this semester, and tomorrow I’ll be talking about that most headache-inducing of problems in probability, the Monty Hall problem.  I hadn’t been planning to go there (because the spring term classes are accelerated, a whole semester’s material in seven weeks, so I keep digressions to a minimum), but now I’ve been asked about it by several of my students. I’m writing this today to get my own thoughts in better order, and hopefully to illuminate the issue for you, dear reader.

If you’ve never heard of the Monty Hall problem (also called the Monty Hall paradox), you’re in for a doozy today. It’s infamous. You can read about it on wikipedia, or here, or here, or here, or in video form here (I could go on almost forever), or even on the website of Let’s Make a Deal (the game show which inspired the present form of the puzzle).  You can even play the game yourself here.

I suppose you could just follow a few of those links and be done with it; and if you don’t like my style then you probably should do just that (but then why are you here?). For my explanation, read on.

The right answer is actually not that complicated, but due to a known bug in human intuition, most people get the wrong answer. Moreover those who get the wrong answer have a tendency to be extremely vociferous in defending their answer. I have actually seen fistfights break out over the answer to this problem. So, discuss this problem with your friends and family only if you’re brave.

(I blog not to unite, but to divide?)

The Monty Hall “paradox” is the story of the following game.  All of my games today involve two characters: Alice (the contestant) and Hatter (the host).

Game 1 Script (Original)

There are three doors.  One leads to a car, and the other two lead to goats.  (The locations of the car and the goats are assigned at random in advance, and Hatter is aware of what is where.)

Alice chooses one door (which she hopes leads to the car) but does not open it.

Hatter, knowing where the car is, opens one of the doors that Alice did not choose, showing her a goat.  (That is, if she picked a goat, he’ll show her the other goat; if she picked the car, he’ll arbitrarily choose a goat to show her.)

Now there are two doors, and Alice knows for sure that one leads to a car and one leads to a goat.  Hatter then offers Alice the option to either switch her choice to the other door or stick with the door she had chosen at the start.

Assuming that Alice wants a car and doesn’t want a goat, is it in her interest to take the prize behind her original door, or is it in her interest to take the prize behind the other door, or does it not matter?

The correct answer is that Alice is twice as likely to get a car if she switches as if she stays; most people think that it shouldn’t matter, that the chances of getting the car are 50-50 either way, so it doesn’t matter.  If that was you, you’re in good company.  Read on for an explanation that will, I hope, be both gentle and convincing.

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