## 25 March 2008

So it's that time of year when March Madness, the NCAA's 65-team college basketball tournament, happens. Lots of betting on this tournament takes place; this usually takes the form of filling out a bracket, and then entering a pool with one's friends where the person who has the "most correct" bracket is the winner. (I suspect most of my US readers are aware of this, but I have lots of non-US readers.) "Most correct" is usually judged by who named the winners of the most games correctly, with some sort of weighting that counts naming the correct winners of later games in the tournament more heavily.

The teams are divided into four groups of sixteen (there are 65 teams, because the two weakest teams in the field play a single game to get things started), and the teams in each group of sixteen are "seeded" 1, 2, ..., 16, with 1 being the perceived best team and 16 being the worst. In the first round of the tournament the #1 team plays the #16 team, the #2 team plays the #15 team, ..., the #8 team plays the #9 team in each group of 16.

One often hears that, say, teams seeded 10 or 11 routinely beat teams seeded 7 or 6, respectively. Fair enough. But therefore it's often claimed that in filling out a bracket one should pick one of those #10 or #11 seeds to win. Not so! Sure, in the average tournament one #10 seed (out of four) might win. But which one?

This is an example of the more general principle that given large enough samples, some rare event will happen... but which one? That's not at all obvious.

Also, Bill James (of baseball analysis fame) wrote an article on when the lead in a college basketball game is safe. Basically, a lead of N points is "safe" if the time in the game remaining is less than kN2 for some constant k; this is what one would expect if scoring can be modeled as a random walk, which seems reasonable.

And you've got to love this quote:
A heuristic could be loosely defined as a mathematical rule that works even though no licensed mathematician would be caught dead associating with it.

As you may have guessed from this post, and the somewhaht desultory nature of the sports content, I'm not really a big basketball fan. But the Red Sox and the A's played a regular-season game this morning in Tokyo. And I've got Phillies tickets for next week.

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