## Roots of Unity

Mathematics: learning it, doing it, celebrating it.

# We Only Need to Fill Out 425 Brackets Each to Win Buffett’s Billion

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Will your bracket be a slam dunk? Image: Acid Pix, via flickr.

Warren Buffett's Bracket Challenge* has put even more of a spotlight than usual on March Madness, the annual NCAA basketball tournament. Buffett has offered a billion dollars to anyone who correctly predicts the outcome of all 63 games in the tournament. There are 2 possible outcomes of every game and therefore 263— 9,223,372,036,854,775,808, or about 9 quintillion—different brackets we could create, giving us a 1 in 9 quintillion chance of winning. Not so hot.

But that estimate assumes that each bracket is equally likely to win, which is clearly false. Even if you know almost nothing about basketball, you're not going to pick a bracket that has the 16 seeds in the Final Four. Jeff Bergen, a math professor at DePaul University, estimates that there is a 1 in 128 billion chance that if you have a good amount of basketball knowledge, you'll pick a correct bracket. Still not great, but much more optimistic than 1 in 9 quintillion. Bergen explained his reasoning in a video he put on YouTube last month.

Bergen's estimates are ballpark figures, based on rough historical averages of how many times each seed has won. His figure of 1 in 128 billion doesn't mean that there is a specific set of 128 billion brackets that definitely contains the winning bracket, but we could use his estimates to figure out which 128 billion brackets are most likely to win. There are about 300 million Americans, so if we managed to make a coordinated effort to keep ourselves from duplicating any brackets, we could each fill out 425 of these likely brackets and be pretty confident that one of us would win! Then we could split the billion dollars 300 million ways and get \$3. Lattes for everyone!

Of course, there's the small caveat that Warren Buffett and Quicken Loans won't let us use this strategy. Unfortunately, the number of entries is capped at 15 million, and each person can only submit one to the official tournament. If we assume each bracket is different, each one is intelligently chosen, and 128 billion is the right number of "intelligent" brackets, and furthermore that they are all equally likely to win (that's a lot of assumptions), there's a little less than a 1 in 10,000 chance that someone wins the billion. Perhaps David Sarno is right in his Slate piece: don't bother filling out a bracket and getting stuck on Quicken's email list.

Bergen's estimates mentioned above don't give any team-specific information on how to pick. They're just based on seed numbers. For more specific recommendations, we'll check out some other mathematical models. Last year, Laura McLay, an operations research professor at the University of Wisconsin wrote a post about some of her favorite ranking tools. This year, Tim Chartier of Davidson College has been all over the place talking about math and bracketology. He and some of his students have gotten very involved in March Madness in the past few years. Some of their best brackets have been above the 99th percentile in ESPN's tournament challenge.

Last Thursday, the Museum of Mathematics hosted a presentation by Chartier about how he harnesses linear algebra to make his predictions (watch a video from his talk here). You can also watch a webinar he gave on bracketology a couple years ago here. And his March MATHness page can help you build a bracket by asking you to make a few choices about how to weight certain aspects of play (schedule, score differential, and so on) and then creating basketball team rankings based on those decisions. If it wins you a billion, you should probably make a donation to Davidson! Just send it to me, and I'll make sure they get it.

*Correction: this post originally misspelled Warren Buffett's surname.

The views expressed are those of the author and are not necessarily those of Scientific American.