This semester I am the teaching assistant for a graduate level course on Statistical Mechanics. To me this represents a milestone. Just a little over two years ago I was in the department's office nearly in tears (okay, actually in tears). I was prepared to quit the program - I was just too stupid for graduate school (I'm still not sure I am smart enough, but that's the subject for another blog). It was only my second week in grad school and already I was convinced that I would never pass Statistical Mechanics.

I don't shed tears over Stat Mech anymore, in fact it's probably one of my favorite classes. Statistical Mechanics is the branch of chemistry (or physics, depending on who you ask) that applies large number statistics to molecules. Since molecules are small and there are so many of them the statistics work out nicely and we can accurately predict thermodynamic quantities (like pressure, energy, entropy, etc).

While brushing up on the subject I was reminded of an interesting statistical problem that I thought I'd share. It's called The Monty Hall Problem, and it's based on the game show "Let's Make a Deal!"

Suppose Monty shows you three doors. You know that behind one of the doors is a new car. Behind each of the other doors is a goat. You choose a door at random (we'll say door #1). Then, Monty opens door #3 and reveals a goat. Monty then makes you an offer - You can switch and take what's behind door #2 instead of door #1. Should you switch or stick with your initial choice?

At first it may seem like switching will make no difference. When the game began your odds of winning a car were 1/3. Monty opens a door and reveals a goat, but that still leaves one goat and one car. The odds must be 50/50, right? The car must be equally likely to be behind either one of the doors. It's often the case, though, that your intuition will deceive you. Already this semester I have warned several students that they were trusting their own intuition a  little too much.

The real answer to the Monty Hall problem is that by switching your choice you move from a 1/3 chance of winning a car to a 2/3 chance. It's important to note that Monty knows where the car is and will never open a door to reveal it (that would ruin the game). Below I outline three ways of convincing yourself that this is the answer. Choose your favorite.

Thinking it through: Making a table
In the beginning of any statistics class you'll get some very easy problems. For example:

"If I roll a 6-sided die¹ what are the chances that I roll a 6?"

These problems are usually pretty easy to answer by just thinking about it or in some cases writing down all the possible outcomes and counting them. Let's write out all the possible outcomes for the Monty Hall problem. This table assumes your choice was door #1 and that Monty will eliminate one of the other doors that has a goat.




Mathematically: Bayes' Theorem
Bayes' Theorem is a statistical tool that lets us analyze the probability of one event happening given that another event has already occurred. In this case, what is the probability that the car is behind door #1, given that Monty reveals a goat behind door #3. The math behind Bayes' theorem is written as: