I’d wondered idly the other day if a number-16 seed had ever lost to a number-one seed in the NCAA Men’s Basketball tournament. This finally made me go and actually try looking it up; a page on statistics.about.com has what it claims are the first-round results from 1985 (when the current 64-team format was adopted) to 2012. This lets us work out roughly the probability of, for example, the number-three seed beating the number-14, at least by what’s termed the “frequentist” interpretation of probability. In that interpretation, the probability of something happening is roughly how many times the thing you’re interested in happens for the number of times it could happen. From 1985 to 2012 each of the various first-round possibilites was played 112 times (28 tournaments with four divisions each); if we make some plausible assumptions about games being independent events (how one seed did last year doesn’t affect how it does this year), we should have a decent rough idea of the probability of each seed winning.
According to its statistics, and remarkable to me, is that apparently the number-one seed has never been beaten by the number-16. I’m surprised; I’d have guessed the bottom team had at least a one percent chance of victory. I’m also surprised that the Internet seems to have only the one page that’s gathered explicitly how often the first rounds go to the various seeds, although perhaps I’m just not searching for the right terms.
From http://bracketodds.cs.illinois.edu I learn that Dr Sheldon Jacobson and Dr Douglas M King of the University of Illinois (Urbana) published an interesting paper “Seeding In The NCAA Men’s Basketball Tournament: When is A Higher Seed Better?” which runs a variety of statistical tests on the outcomes of March Madness tournaments and finds that the seeding does seem to correspond to the stronger team in the first few rounds, but that after the Elite Eight round there’s not the evidence that a higher seed is more likely to win than the lower; effectively, after the first few rounds you might as well make a random pick.
Jacobson and King, along with Dr Alexander Nikolaev at SUNY/Buffalo and Dr Adrian J Lee, Central Illinois Technology and Education Research Institute, also wrote “Seed Distributions for the NCAA Men’s Basketball Tournament” which tries to model the tournament’s outcomes as random variables, and compares how these random-variable projections compare to what actually happened between 1985 and 2010. This includes some interesting projections about how often we might expect the various seeds to appear in the Sweet Sixteen, Elite Eight, or Final Four. It brings out some surprises — which make sense when you look back at the brackets — such as that the number-eight or number-nine seed has a worse chance of getting to the Sweet Sixteen than the eleventh- or twelfth-seed does.
(The eighth or ninth seed, if they win, have to play whoever wins the sixteen-versus-one contest, which will be the number-one seed. The eleventh seed has to beat first the number-six seed, and then either the number-three or the number-14 seed, either one of which is more likely.)
Meanwhile, it turns out that in my brackets I had picked Connecticut to beat Villanova, which has me doing well in my group — we get bonus points for calling upsets — apart from the accusations of witchcraft.