Math concepts used: Covariance and correlations. The normal distribution and the central limit theorem.
Background: On December 5, 2014, in front of a packed house at Gampel Pavilion,
the Yale men's basketball team pulled off a stunning upset over the UConn men's basketball team, 45-44. (Watch
Jack Montague's game-winning three-pointer in the same link.) The last
time Yale won over UConn dates back to December 2, 1986.
Also of note: Five days (the Sunday) prior, UConn lost to Texas on a last-second three-pointer launched from almost the exact same spot (left corner baseline) where Montague did.
Relevance: I was in attendance at the game, wearing both a Yale sweater and a UConn sleeved shirt. Believe me, I was quite torn between the two sides! But as soon as I saw that UConn trailed with 5 minutes left in the game, I knew that history was about to happen... [Disclosure: I watched the previous two contests between UConn and Yale on TV, in 2003-04 and 2013-14. Both years UConn went on to win the national title.]
What I did: The game took place on the last day of classes for the semester; final exams started the following Monday. After the game, in consultation with a colleague, I determined that a final exam problem was in order. [I had warned my students in advance that such a problem might appear on the exam.] So over the course of the weekend (December 6--7, 2014), I looked up past matchup stats between UConn MBB and Ivy League opponents on Sports Reference, starting from 1980. Since UConn played on average one Ivy League opponent a year (need not be Yale, could be Harvard), and team performance varied widely from year to year, I made a facile assumption that each matchup was statistically independent. Then I took note the margin of error from each matchup, calculated the mean and the standard deviation, and obtained an approximate probability that UConn wins over (or loses to) an Ivy League opponent. This then served as the basis for the actual exam problem.
Aftermath: I tweeted out to @NoEscalators, who then retweeted it. Soon afterwards
Matt Norlander of CBSSports.com contacted me to do
a quick write-up on this entire episode. I obliged, and the rest is history. And lots thereof.
Here is Matt Norlander's write-up.
Please don't mind the typos in the article. I was in a hurry feeding him the background info (having to catch a plane
the next day), and to his credit, he wrote a story which was pretty faithful to my thought process.
Several news and sports outlets reposted or followed up on Norlander's story. There are too many to count, so I will just mention that Keith Olbermann, my fellow
Cornellian, called me out on his erstwhile eponymous TV program as a "Worse Person of the Day." I disagree with that moniker, okay?
I also want to thank many people on The Boneyard (a forum for UConn sports), including a student of mine, for defending my reputation against a poster who
wanted to put my head on a pike.
Of course I would be remiss not to thank the support from my alma mater. Thanks to the Yale Alumni Magazine and the Yale basketball program
for the social media mentions. And as much as this might have upset my employer, the UConn College of Liberal Arts & Sciences (CLAS) was gracious enough to put up a blurb on my behalf.
In fact, the following semester, I was introduced in a class taught by the CLAS Dean, Jeremy Teitelbaum (himself a mathematician), as the "person who got famous for writing the UConn basketball problem." He
then relayed to me a story about mathematicians at a university (which shall remain nameless) who played a "rank U.S. math grad programs" NCAA-tournament-style.
Aftermath, the mathematicians' version:
I have also received several reactions here and abroad with regard to the math. To clarify the main arguments, I am listing them FAQ style.
Question: (From several commenters, including someone working in Europe.) Why did you give the hint "122=144 and 152=225"? Don't your students know how to do those by hand?
Answer: Yes, I am confident that most of my students can do the arithmetic by hand, without the help of calculators (which are banned during my exams). The point
of the problem is about stringing together several probability concepts, and not so much about crunching numbers.
This goes along with my primary philosophy in teaching mathematics: students are here to learn the big ideas and the important concepts behind a mathematical theory. Computation
is important, but if one cannot even grasp the concepts behind a word problem, I don't care how many calculator key punches one makes, it's not going to help solve the problem.
Therefore I NEVER emphasize numerical calculations in my exams. If I see that the student makes all the correct reasoning but messes up just a bit in the numerics, I'm happy to give
18~19 points out of 20. But if I see some faulty work to begin with (whether the numerical answer is right or wrong), that indicates a lack of conceptual understanding.
Question: (From a commenter on CBSSports.com, suitably paraphrased.) The central limit theorem does not apply in this case.
Answer: One may criticize that my sample size (32) isn't large enough. That is fine, I could have dug up more data if I had time (more than a weekend). But I would say that the problem
fits the criteria of the central limit theorem pretty well. The games were usually played in distinct seasons, and with the ever-changing roster, it is safe to assume that they were independent.
No one knows what distribution these stats were drawn from, so one assumes some underlying "black box" distribution. It certainly has finite 2nd moment (the number of samples is finite after all). That is
all you need to use the central limit theorem. Did I miss anything else? (Probabilists, please tell me that I should use the Berry-Esseen theorem to characterize the error in the CLT approximation!)
Question: (From another commenter on CBSSports.com.) I counted the number of wins and losses (read: used the uniform distribution) and found that the probability of UConn being upset was 6%, which was not far from your more elaborate
method of using the margin of error to get the 7%. So why bother using your method?
Answer: I believe that the closeness of the numbers is coincidental. I do not dispute that the uniform distribution might be a good approximation in the event studied here (UConn is upset by the opponent).
However, try estimating the probability of a different (and more probable) event, say, UConn beats the opponent by more than 10 points. The central limit theorem will still work, but I'm not sure about using the uniform assumption.
Acknowledgements: Many thanks to Keith Conrad for joining me for the historic game. (We two share the dubious distinction of being among the <1% of UConn affiliates having any rooting interest for Yale in Gampel that night.) He also helped check the wording of the problem before it went into the exam, and relayed the aforementioned comments from a commenter across the Atlantic.