# Final Four… 1 in 4? from Alston Mabry

April 3, 2010 |

By the time the NCAA Men's Division I basketball tournament gets down to four teams, is it a toss-up as to which team wins? In other words, does each team have a 1 in 4 chance of winning?

## Steve Ellison responds:

This is an excellent situation to apply the binomial theorem. In E****, you can write a formula: =BINOMDIST(s,t,p,c), where s is the number of successes, t is the number of trials, p is the probability of success, and c indicates whether to calculate the cumulative result (1=yes, 0=no)

Our null hypothesis is that the probability of success (winning the Final Four) is 0.25.

For example, for the teams that have made only one trip to the Final Four, the formula is =BINOMDIST(1,20,0.25,1), resulting in a p value of 0.024. This result appears to be statistically significant, but the significance is questionable given that we are doing multiple comparisons.

Teams that have made at least four trips to the Final Four have won more than a quarter of the time, so we can check the probability of winning 21 times or less and subtract it from 1, using the formula =1-BINOMDIST(21,65,0.25,1), which results in a p value of 0.070.

This is retrospective, and has "survivorship bias" or winner's bias.

This would be true if the championship wins and returning to the Final Four where independent. But they are not. If you win one year, the record most likely show you have a better chance of repeating in the final four.

You need stats that show winner only in their first appearance or second appearance, etc.

Here are the Final Four champs by appearance order:

appearance  count champs  binomdist

1st                  97     13          0.4%
2nd                 55     15         71.3%
3rd                 32      8          59.4%
4th                 24      9          94.5%
5th or more      80     25         92.0%
4th or more     104    34         97.0%

Looks like an opportunity to run a quick sim. The sim sets up the Final Four participants for each year 1979-2009, randomly assigns the National Championship to one of the schools, and then records whether that school was one of the schools that actually went to the Final Four just once, twice, three times, or four or more times. Results of 1000 runs:

schools that went to the Final Four just once
count: 20
actual championships: 1
mean # of championships in 1000 sim runs: 5.00
sd of this distribution: 1.68
z score of actual # of championships compared to sim distribution: -2.38

schools that went to the Final Four exactly twice
count: 9
actual championships: 2
mean # of championships in 1000 sim runs: 4.49
sd of this distribution: 1.73
z score of actual # of championships compared to sim distribution: -1.44

schools that went to the Final Four exactly three times
count: 7
actual championships: 6
mean # of championships in 1000 sim runs: 5.33
sd of this distribution: 1.92
z score of actual # of championships compared to sim distribution: +0.35

schools that went to the Final Four four times or more
count: 11
actual championships: 22
mean # of championships in 1000 sim runs: 16.19
sd of this distribution: 2.53
z score of actual # of championships compared to sim distribution: +2.30

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