Final Four Probabilities

For the first time in history, the teams in the Final Four this year are all number 1 seeds. I've been hearing about this for a while, and began to wonder... Should my reaction be "it's about time" or "wow, that's remarkable"? So I went after the numbers to try to figure it out.

The first thing you realize is that the NCAA likes to change formats pretty regularly. The current format (dating back to 2002) starts with 65 teams; the previous 16 seasons featured a 64-team tournament with the same overall structure. These (1985 - 2007) are the tournaments whose results I used. Here's what I found:

Seed	Region Wins	Percent
1	38	0.41
2	21	0.23
3	12	0.13
4	9	0.1
5	4	0.04
6	3	0.03
8	3	0.03
11	2	0.02

So, if we take history as a guide, we would put the probability of all four #1 seeds winning their regional brackets at just under 3% [.41 * .41 * .41 * .41], so it should happen once every 34 years, on average. This is the 24th year of the current format, so it happened a little sooner than we would have expected.

An interesting question is, "what is the most likely distribution of seeds in a Final Four?" Obviously, a 1 seed is the favorite to win its regional group, but other outcomes are more likely than all of them winning. Three 1 seeds and a 2 seed would be twice as likely (about 6.4%, once every 15.5 years); it happened in 1993. Close behind that, we would expect two 1 seeds, a 2 seed and a 3 seed about every 16.4 years; this was the case in 1991 and 2001.

Two years ago, no number one seeds made the Final Four (this was not quite a first - it happened in 1980 during a 48-team tournament). This is actually a relatively likely situation - you would expect it once every 12 or so years.

To close out, let's look at outcomes that we would expect more frequently, and see how they compare with history:

Final Four Seeds	Probability	Expected Time (Years)	Occurrences	Observed Rate
3 #1's	0.1654	6.04	3	0.13
2 #1's	0.3527	2.84	10	0.43
1 #1	0.3341	2.99	9	0.39

Note that the above outcomes are mutually exclusive (but not exhaustive - I mentioned the other cases above). In particular, I'm surprised by the frequency with which only one 1 seed makes it to the Final Four. I wonder what kind of odds I can get on that next year...