Warren Buffett and Quicken Loans made quite a splash last week when they announced that they would be awarding $1 billion to anyone who filled out a perfect bracket for this year’s NCAA Men’s Division I Basketball Tournament.
As many news articles all over the web have already pointed out, the total number of possible variations is enormous. In a single-elimination bracket involving 64 teams, there are exactly 63 matches played. As each match can have two outcomes, the total number of possible outcomes is 2^63, or 9,223,372,036,854,780,000 (roughly 9.2 quintillion).
However, without knowing too much about basketball, much less having an educated view of which teams are most likely to make the tournament this year, we can actually narrow that number down by quite a bit.
Part A: The first and most important step is noting that in the history of the tournament, no #1 seed has ever lost to a #16 seed in the first round (116-0). Assuming things stay that way this year, we can reduce the number of variations to 2^59 (since 4 of the matches are decided), which is 576,460,752,303,423,000. This is a still a massive number, but only 6.25% of the theoretical total.
Part B: Continuing off of A, we can build on the concept of upsets to further narrow the number of bracket variations. If we define an upset as simply any result where a lower ranked seed defeats a higher ranked seed, then the result of every match becomes binomial: it is either an upset, or it isn’t. This is a powerful notion, because we can now think of the total number of variations as the sum of distinct sub-variations based on the number of upsets. In a tournament consisting of 63 matches, there can either be 0 upsets, 1 upset, 2 upsets, 3 upsets…62 upsets, or 63 upsets. Furthermore, there is (63 choose 0)=1 way of achieving 0 upsets, (63 choose 1)=63 ways of achieving 1 upset, etc. Summing up all of these terms in fact gives us 9,223,372,036,854,780,000, which is what we expect. Astute readers will note that this is simply a combinatorial identity demonstrated by Pascal’s Triangle, but it is nonetheless meaningful to take a moment to verify this identity yourself if you’ve never encountered it. While it is theoretically possible for any number of upsets to occur during a particular tournament, the last five NCAA tournaments have all had between 16 and 20 upsets (average of 18.4). Thus, if we only consider brackets with 15 to 22 upsets (going out by ~2 standard deviations on both ends for some margin of safety), the number of variations drops down to 19,420,762,596,874,200, or 0.211% of the theoretical total.
Part C: We make one last simple observation to refine our analysis: for any given tournament, most of the upsets (about 50%) occur in the first round (average of 9.2), with no less than 7 upsets in the first round over the last five years. Thus if we disregard all the bracket variations in B that include 5 upsets or less in the first round, we can cut down the number of variation by an additional 200 trillion of so, giving us 19,219,810,265,601,500 variations, or 0.208% of the theoretical total.
Conclusion: Clearly, the odds of predicting a perfect bracket is still extremely small. However, the key takeaway here is that even with no knowledge of college basketball, we can quickly reduce the total theoretical number of bracket variations by almost three orders of magnitude to a more practical number by using basic probability concepts and making a few simple assumptions.
P.S. While you do have a better chance of winning the lottery than getting your hands on the grand prize of 1 billion dollars, the NPV of this particular “investment” is still positive since the cost of registration is free 😛
