## Saturday, March 06, 2004

### And Then There Was One

With Stanford's loss to Washington this afternoon, there is only one undefeated team left in the men's NCAA Division I: the St. Joseph's Hawks, who are currently 27-0, with the Atlantic 10 Conference Tournament (held in the coastal city of Dayton, Ohio) to go. No D-I team has finished a perfect regular season plus conference tournament since UNLV in 1990-91, and no team has had a perfect Division I season since Indiana did it in 1975-76. Only UNLV and Indian State (1979) have gotten to the NCAA championship game with a perfect record, and they both lost.

Since none other than Roy Williams has said that the NCAA tournament is a crapshoot, I got to wondering what the odds were that we'd see a perfect season anytime soon. Since a team with a perfect record is likely to be the number 1 seed, I decided to look at all of the number 1 seeds in the 64-team tournament era (1985 on) to calculate a lower bound on the probability. I got most of my data from The Official 2002 NCAA Men's Final Four Tournament Records Book, and I looked up individual team records for 2003.

Start with the regular season, which, to the NCAA, includes the conference tournaments, if any. Since the 1984-5 season, the 76 teams with Number 1 seeds have gone 2086-266. In other words, an average Number 1 seed has a 0.8869048:1 chance of winning any given regular season game. Since a typical regular season is 30 games (including the conference tournament), this means that the probability than an average Number 1 seed (hereafter AN1S) has a 0.8869048^3:1 = 0.0273092:1 chance of a perfect regular season. Thus, out of the 76 N1S since 1985, we'd expect 0.027*76 = 2 teams to have a perfect record through the regular season. We've had 1 (UNLV, 1991), so our estimate is reasonable.

Now for the tournament. To finish the perfect season, a team must win 6 tournament games. Of the 19 NCAA championships in the 64-team era, 11 have been won by a N1S. Thus the probability of a given N1S winning the tournament is 11/76 = 0.1447368:1.

So the probability of a N1S having a perfect regular season record is 0.0273092:1, and the probability of a N1S winning the tournament is 0.1447368:1, then the probability that a team will finish the season with a perfect record is 0.0273092*0.1447368 = 0.0039527:1. With 76 teams having had a shot at this, we'd expect 0.00395*76 = 0.300 teams with a perfect record. OK, we don't have any, so this isn't all that bad an estimate.

But 0.3 isn't all that small either. With these odds, we could have had a perfect season in the last 19 years, just taking an AN1S and a little bit of luck. So we could have a perfect season this year. St. Joe's? Well, they're the only one left. I wouldn't bet on them winning, but it's not all that impossible.

And Roy? A N1S has one chance out of seven of winning it all. You've had 5 chances. The clock is ticking.