Now that the bracket for the 2013 NCAA Men's Division I Basketball Tournament has been decided, millions of people will attempt to correctly predict the outcome of the tournament in advance. While it is important to consider factors such as the recent record of each team, the schedule strength of each team, how players on competing teams match up against each other, and how deep each team's talent pool is, this guide will examine the tournament with a purely mathematical approach, using probability and statistics to determine how many upsets of each kind should be chosen during the Round of 64. All historical considerations for this guide begin with the 1985 NCAA Men's Division I Basketball Tournament, as it was the first to include at least 64 teams.
Methodology
The statistical formula for predicting the odds of a given number of upsets of a particular type is
U^m*(1-U)^n*(m+n)! , (1)
m!*n!
where U is the odds of an upset, m is the number of upsets of that type, and n is the number of non-upsets of that type. In the Round of 64, m+n=4 in every case, as there are always four matchups of each type. The factorials in the formula account for the combinatorics of the arrangements of the upsets.
#1 seed vs. #16 seed
In 112 games since 1985, #1 seeds are 112–0 against #16 seeds. While some #16 seeds have come close to an upset, it is generally unwise to predict that something unlikely and historic will occur. The 112–0 record would suggest that the odds of a particular #16 seed upsetting the #1 seed it is paired with are less than or equal to (1/112)*ln(2), or 0.619%. Thus U≤0.00619. Using Equation (1), the odds are as follows:
- All #1 seeds advance: ≥97.55%
- One #16 seed upsets a #1 seed: ≤2.43%
- Two #16 seeds upset #1 seeds: ≤0.023%
- Three #16 seeds upset #1 seeds: ≤9.4*10^-5 %
- All #16 seeds advance: ≤1.5*10^-7 %
Mathematically, all #1 seeds should be predicted to advance. An upset pick of this type would not be reasonable.
#2 seed vs. #15 seed
In 112 games since 1985, #2 seeds are 106–6 against #15 seeds. This is an upset rate of 5.36%, thus U=0.0536. The 2012 tournament was notable for being the first time that more than one such upset occurred in the same year. Using Equation (1), the odds are as follows:
- All #2 seeds advance: 80.22%
- One #15 seed upsets a #2 seed: 18.17%
- Two #15 seeds upset #2 seeds: 1.54%
- Three #15 seeds upset #2 seeds: 0.058%
- All #15 seeds advance: 8.3*10^-4 %
Mathematically, all #2 seeds should be predicted to advance. An upset pick of this type would not be reasonable.
#3 seed vs. #14 seed
In 112 games since 1985, #3 seeds are 96–16 against #14 seeds. This is an upset rate of 14.29%, thus U=0.1429. The most such upsets that have occurred in one year is two, in 1986 and 1995. Using Equation (1), the odds are as follows:
- All #3 seeds advance: 53.97%
- One #14 seed upsets a #3 seed: 35.99%
- Two #14 seeds upset #3 seeds: 9.00%
- Three #14 seeds upset #3 seeds: 1.00%
- All #14 seeds advance: 0.042%
Mathematically, all #3 seeds should be predicted to advance, but one upset pick of this type would also be reasonable.
#4 seed vs. #13 seed
In 112 games since 1985, #4 seeds are 88–24 against #13 seeds. This is an upset rate of 21.43%, thus U=0.2143. The most such upsets that have occurred in one year is two, in 1987, 2001, and 2008. Using Equation (1), the odds are as follows:
- All #4 seeds advance: 38.11%
- One #13 seed upsets a #4 seed: 41.58%
- Two #13 seeds upset #4 seeds: 17.01%
- Three #13 seeds upset #4 seeds: 3.09%
- All #13 seeds advance: 0.21%
Mathematically, one #4 seed should fall to a #13 seed, but no upset pick of this type would also be reasonable.
#5 seed vs. #12 seed
In 112 games since 1985, #5 seeds are 74–38 against #12 seeds. This is an upset rate of 33.93%, thus U=0.3393. The most such upsets that have occurred in one year is three, in 2002 and 2009. Using Equation (1), the odds are as follows:
- All #5 seeds advance: 19.06%
- One #12 seed upsets a #5 seed: 39.14%
- Two #12 seeds upset #5 seeds: 30.15%
- Three #12 seeds upset #5 seeds: 10.32%
- All #12 seeds advance: 1.33%
Mathematically, one #5 seed should fall to a #12 seed, but two upset picks of this type would also be reasonable.
#6 seed vs. #11 seed
In 112 games since 1985, #6 seeds are 74–38 against #11 seeds. This is an upset rate of 33.93%, thus U=0.3393. A complete ousting of #6 seeds by #11 seeds occurred in 1989. Using Equation (1), the odds are as follows:
- All #6 seeds advance: 19.06%
- One #11 seed upsets a #6 seed: 39.14%
- Two #11 seeds upset #6 seeds: 30.15%
- Three #11 seeds upset #6 seeds: 10.32%
- All #11 seeds advance: 1.33%
Mathematically, one #6 seed should fall to a #11 seed, but two upset picks of this type would also be reasonable.
#7 seed vs. #10 seed
In 112 games since 1985, #7 seeds are 67–45 against #10 seeds. This is an upset rate of 40.18%, thus U=0.4018. A complete ousting of #7 seeds by #10 seeds occurred in 1999. Using Equation (1), the odds are as follows:
- All #7 seeds advance: 12.81%
- One #10 seed upsets a #7 seed: 34.40%
- Two #10 seeds upset #7 seeds: 34.66%
- Three #10 seeds upset #7 seeds: 15.52%
- All #10 seeds advance: 2.61%
Mathematically, two #7 seeds should fall to #10 seeds, but one upset pick of this type would also be reasonable.
#8 seed vs. #9 seed
In 112 games since 1985, #8 seeds are 54–58 against #9 seeds. This is an upset rate of 51.79%, thus U=0.5179. A complete ousting of #8 seeds by #9 seeds occurred in 1989, 1994, 1999, and 2001. Using Equation (1), the odds are as follows:
- All #8 seeds advance: 5.40%
- One #9 seed upsets a #8 seed: 23.21%
- Two #9 seeds upset #8 seeds: 37.40%
- Three #9 seeds upset #8 seeds: 26.79%
- All #9 seeds advance: 7.19%
Mathematically, two #8 seeds should fall to #9 seeds.
Overall
Since 1985, the Round of 64 has featured a mean of 8.04 upsets, a median of 8 upsets, and a mode of 9 upsets. The fewest upsets in the Round of 64 was 3 (2000), and the most upsets was 13 (2001).
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