Now that the bracket for the 2013 NCAA Women'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 1994 NCAA Women'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 76 games since 1994, #1 seeds are 75–1 against #16 seeds. This is an upset rate of 1.32%, thus U=0.0132. The only time such an upset occurred was in 1998. Using Equation (1), the odds are as follows:
- All #1 seeds advance: 94.82%
- One #16 seed upsets a #1 seed: 5.07%
- Two #16 seeds upset #1 seeds: 0.10%
- Three #16 seeds upset #1 seeds: 9.1*10^-4 %
- All #16 seeds advance: 3.0*10^-6 %
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 76 games since 1994, #2 seeds are 76–0 against #15 seeds. While some #15 seeds have come close to an upset, it is generally unwise to predict that something unlikely and historic will occur. The 76–0 record would suggest that the odds of a particular #15 seed upsetting the #2 seed it is paired with are less than or equal to (1/76)*ln(2), or 0.912%. Thus U≤0.00912. Using Equation (1), the odds are as follows:
- All #2 seeds advance: ≥96.40%
- One #15 seed upsets a #2 seed: ≤3.55%
- Two #15 seeds upset #2 seeds: ≤0.049%
- Three #15 seeds upset #2 seeds: ≤3.0*10^-4 %
- All #15 seeds advance: ≤6.9*10^-7 %
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 76 games since 1994, #3 seeds are 76–0 against #14 seeds. While some #14 seeds have come close to an upset, it is generally unwise to predict that something unlikely and historic will occur. The 76–0 record would suggest that the odds of a particular #14 seed upsetting the #3 seed it is paired with are less than or equal to (1/76)*ln(2), or 0.912%. Thus U≤0.00912. Using Equation (1), the odds are as follows:
- All #3 seeds advance: ≥96.40%
- One #14 seed upsets a #3 seed: ≤3.55%
- Two #14 seeds upset #3 seeds: ≤0.049%
- Three #14 seeds upset #3 seeds: ≤3.0*10^-4 %
- All #14 seeds advance: ≤6.9*10^-7 %
Mathematically, all #3 seeds should be predicted to advance. An upset pick of this type would not be reasonable.
#4 seed vs. #13 seed
In 76 games since 1994, #4 seeds are 70–6 against #13 seeds. This is an upset rate of 7.89%, thus U=0.0789. There has never been more than one upset of this type in the same year. Using Equation (1), the odds are as follows:
- All #4 seeds advance: 71.98%
- One #13 seed upsets a #4 seed: 24.66%
- Two #13 seeds upset #4 seeds: 3.17%
- Three #13 seeds upset #4 seeds: 0.18%
- All #13 seeds advance: 3.9*10^-3 %
Mathematically, all #4 seeds should be predicted to advance. An upset pick of this type would not be reasonable.
#5 seed vs. #12 seed
In 76 games since 1994, #5 seeds are 60–16 against #12 seeds. This is an upset rate of 21.05%, thus U=0.2105. The most such upsets that have occurred in one year is two, in 1996, 1998, 2002, and 2009. Using Equation (1), the odds are as follows:
- All #5 seeds advance: 38.85%
- One #12 seed upsets a #5 seed: 41.44%
- Two #12 seeds upset #5 seeds: 16.57%
- Three #12 seeds upset #5 seeds: 2.95%
- All #12 seeds advance: 0.20%
Mathematically, one #5 seed should fall to a #12 seed, but no upset picks of this type would also be reasonable.
#6 seed vs. #11 seed
In 76 games since 1994, #6 seeds are 53–23 against #11 seeds. This is an upset rate of 30.26%, thus U=0.3026. The most such upsets that have occurred in one year is three, in 2006. Using Equation (1), the odds are as follows:
- All #6 seeds advance: 23.66%
- One #11 seed upsets a #6 seed: 41.06%
- Two #11 seeds upset #6 seeds: 26.72%
- Three #11 seeds upset #6 seeds: 7.73%
- All #11 seeds advance: 0.84%
Mathematically, one #6 seed should fall to a #11 seed.
#7 seed vs. #10 seed
In 76 games since 1994, #7 seeds are 51–25 against #10 seeds. This is an upset rate of 32.89%, thus U=0.3289. The most such upsets that have occurred in one year is two, in 1994, 1995, 1996, 1998, 1999, 2000, 2001, 2009, and 2011. Using Equation (1), the odds are as follows:
- All #7 seeds advance: 20.28%
- One #10 seed upsets a #7 seed: 39.76%
- Two #10 seeds upset #7 seeds: 29.23%
- Three #10 seeds upset #7 seeds: 9.55%
- All #10 seeds advance: 1.17%
Mathematically, one #7 seed should fall to a #10 seed.
#8 seed vs. #9 seed
In 76 games since 1994, #8 seeds are 37–39 against #9 seeds. This is an upset rate of 51.32%, thus U=0.5132. The most such upsets that have occurred in one year is three, in 1994, 1996, 1998, 2000, 2001, 2002, 2009, and 2011. Using Equation (1), the odds are as follows:
- All #8 seeds advance: 5.62%
- One #9 seed upsets a #8 seed: 23.68%
- Two #9 seeds upset #8 seeds: 37.45%
- Three #9 seeds upset #8 seeds: 26.32%
- All #9 seeds advance: 6.94%
Mathematically, two #8 seeds should fall to #9 seeds.
Overall
Since 1994, the Round of 64 has featured a mean of 5.79 upsets, a median of 6 upsets, and a mode of 6 upsets. The fewest upsets in the Round of 64 was 2 (1997 and 2003), and the most upsets was 10 (1998).
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