The Monty Hall problem

On a plane flight back from a recent trip, I watched the movie 21. The plot, advertised as “based on a true story”, is roughly as follows. (In case you have not seen the movie and would like to, I will try to avoid major spoilers.)

Ben Campbell, an idealistic and somewhat naive MIT student, impresses a math professor (Kevin Spacey) by answering correctly a couple of tricky questions in class. Soon afterwards, the professor asks Ben to join his card-counting blackjack team in return for a share of the profits. The team travels to Las Vegas on weekends, plays blackjack at major casinos, and wins millions of dollars by placing themselves strategically at the right tables and employing the card-counting techniques taught by the professor. Ben refuses at first (“and if you tell anybody, I’ll make sure that you won’t graduate”), but there’s no other way that he can pay for his dream med school, and he’s attracted to a female student on the team, and besides, if he didn’t join the team, there would be no movie, so guess what happens.

That’s about the first quarter of the movie, and I’ll leave it there, because this is already enough to raise serious questions about just how close to a “true story” we are here.

My first question was, has there really been an MIT math professor who made a fortune off a team of student card players? Wouldn’t that be serious professional misconduct, and would an MIT professor (not a bad job) really take this sort of risk? As it turns out, the movie is somewhat loosely based on the adventures of a real-life MIT card counting team (one of several that MIT has had over the years). However, the teams were all entirely composed of, and run by, the students. There were no professors involved, and the Spacey character is completely fictional.

Which also preempted my follow-up question. There’s a classroom scene where the Spacey character asks his students what applications of Newton’s method they know. A student suggests, “Nonlinear equations?…” Spacey responds along the lines of “Yeah, that’s very clever, because this course is called Nonlinear Equations. Why don’t you tell me something I don’t already know.” My impression was that Spacey’s demeanor, and this exchange in particular, was a little bit too snarky. A professor is not supposed to do that if he (or, especially, she) wants to get good student evaluations.

The user comments on IMDB include several reviews by authors who appear to be well familiar with casinos, blackjack and card counting, and are not entirely happy with the treatment of the subject. I’ve never played blackjack, or been to a casino, but their criticism makes sense to me.

But here’s the reason why I’m writing this post. How exactly does Ben manage to impress Spacey’s character? Spacey asks him the following question:

You’re on a game show. The host asks you to choose one of three doors. Behind one of them there’s a car, behind each of the other two there’s a goat. You pick one door, say Door 1. The host, who knows where the car is and is not allowed to reveal that information, opens Door 3, behind which there is a goat. He then asks you if you want to choose Door 2 instead of 1. Is it to your advantage to switch?

In case you’d like to think about it, the rest of post is behind the cut.

Yes, you should switch to Door 2. Why? The short version is as follows. Initially, each door is the winning one with probability 1/3. If you stay with Door 1, you win if and only if your initial choice was the winning door, an event with probability 1/3. If you switch to Door 2, you win if and only if your initial choice was not the winning door, an event with probability 2/3. Therefore you are more likely to win if you switch. That’s the answer Ben gave (though I didn’t think that either he or Spacey explained it very clearly).

More details here, including a discussion of just how the details of the set-up can make all the difference. Enjoy!

1 Comment

Filed under mathematics: general, movies

One response to “The Monty Hall problem

  1. liuxiaochuan

    Dear Professor Laba:

    I myself am a huge movie fan. About a while ago I saw this movie and didnot get impressed in any way. I think the plot is kind of stereotypical and my guess about the end is correct.

    I already heard about this probabilistic question before this movie. As I remenbered, it is raised by some professor and once aroused a controversy because of the unclear statement about the initial conditions. In this movie, Spacey character emphasized the host knows the result which made the difference.

    Any way, It is always a pleasure to see that there are people who make movies about mathematics.