The Mathematical Madness Behind a Perfect N.C.A.A. Basketball Bracket

The probability of picking a perfect NCAA March Madness bracket is astronomically low. If college basketball players were immortal beings that sprung into existence at the moment of the Big Bang, and they competed in the 64-team NCAA basketball tournament every year for the 13.8-billion-year history of the universe, and someone filled out a tournament bracket randomly each year, they still, almost certainly, wouldn’t pick a perfect bracket.

Such are the numbers of March Madness, the annual tradition of guessing the outcome of 63 basketball games in a single-elimination tournament, an impossible task that President Barack Obama called “a national pastime.” The probability of a perfect bracket is so low that Warren Buffet offered a billion dollars to anyone who could pull it off in 2014 (no one did, or ever has, as far as we know). Even so, every year statisticians and computer scientists crunch the numbers to try to produce the closest bracket to perfection among tens of millions that are filled out each year, knowing that picking every game correctly is beyond the capability of mere mortals.

“I don’t think there’s anything that captures the social consciousness’ attention [as much] as March Madness,” says Tim Chartier, an applied mathematics and computer science professor at Davidson College who specializes in sports analytics. “There’s something alluring about the whole thing in that [the bracket, inevitably,] does get busted.”

If you were to pick randomly, the probability of picking a perfect March Madness bracket is 1 in 2⁶³, or about 1 in 9.2 quintillion. You have a better chance of winning Powerball twice in a row, or getting hit with a piece of space junk falling out of the sky.

You can improve your bracket with knowledge of the sport, but to what extent is a matter of debate. For example, most March Madness players consider it a safe bet to choose all the No. 1 seed teams to win their first-round matchups against No. 16 seed teams, considering a No. 1 seed had never lost to a No. 16 seed until the University of Maryland, Baltimore County, upset the University of Virginia last year. (Top seeded teams have won 135 of 136 games over No. 16 seed teams since the modern tournament began in 1985.)

“The simplest thing is to ask yourself is, how many games of the 63 are you willing to say, ‘I will have 100 percent chance of winning,’” says Mark Ablowitz, an applied mathematics professor at the University of Colorado, Boulder.

If all the No. 1 seeds were guaranteed to win their first-round games, and every other game were chosen at random, the probability of a perfect bracket would improve to 1 in 2⁵⁹, or about 1 in 576 quadrillion compared to 9.2 quintillion. Of course, the No. 1 seeds are not guaranteed to win in the first round, so we may say that the probability—assuming you pick all No. 1 seeds in the first round—is somewhere between 1 in 576 quadrillion and 1 in 9.2 quintillion.

So how far can knowledge of the sport take you? For every game you can reliably pick correctly, the probability of a perfect bracket improves exponentially. Could you incorporate enough information into the decision-making process to bring a perfect bracket into the realm of statistical possibility?

Bracketology

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Chartier leads a group of student researchers every year who test mathematical methods of picking teams in March Madness. “It gets people thinking math and thinking statistics but also seeing the uncertainty of the whole thing,” he says.

His basic method is simple, weighting the teams based on variables other than their regular season records. “One of the worst brackets you can make is just solely based off winning percentage,” Chartier says. Instead, a statistical method might weight the teams’ rankings based on when the games were played, the challenge of opponents and the number of points each game was won or lost by.

For example, you might take all the games in the first half of the regular season and weight them so a win is only worth half a win and a loss is worth half a loss. “That way, I’m saying that the games in the second half [of the season] are more predictive of winning in March Madness.”

Using such methods, Chartier and his students frequently produce brackets within the 97th percentile of the millions of brackets submitted annually to ESPN’s online “Tournament Challenge.” The students are encouraged to tweak the weighting method, or consider additional variables when games are predicted to be close in the baseline analytics. One year, a student of Chartier scored within the 99.9th percentile of brackets submitted to ESPN. When Chartier reviewed her method to see what she had done, he found that she factored in home and away games, weighting away game wins as a better indicator of winning in March Madness than home game wins. Chartier now includes home and away data in his method as well.

Exactly what variables to consider, however, isn’t always clear. In 2011, neither a No. 1 seed nor a No. 2 seed made it to the Final Four for the first time in tournament history. Butler, a No. 8 seed, made a run all the way to the finals that few sports fans or statisticians predicted. Chartier didn’t predict Butler’s run, but one of his students did by incorporating regular season winning streaks into her weighting system.

In 2008, No. 10 seed Davidson, with future NBA superstar Steph Curry, made an unexpected run to the Elite Eight. Chartier teaches at Davidson, but even so, “we have not been able to produce methods that predict that they did so well,” he says.

In the future, Chartier hopes to incorporate the experience of players and coaches as well as the impact of injuries on regular season wins and loses into his method, but he hasn’t yet found a good statistical way to do so. “If we can’t do it for all the teams, then we don’t do it,” he says.

But there is a big difference between picking games better than most people and picking a perfect bracket. When it comes to the probability of selecting a perfect bracket, nobody knows for sure. Chartier says that historically, researchers using statistical methods have reliably picked about 70 percent of the games correctly, making the probability of a perfect bracket (assuming you can choose correctly 70 percent of the time) 1 in 1/.70⁶³, or about 1 in 5.7 billion. If you could improve your winning percentage to 71 percent, the probability of a perfect bracket improves to 1 in 2.3 billion, and if you could reliably pick the winner of each game 75 percent of the time, the probability of perfection jumps all the way to 1 in 74 million.

Unfortunately, things might not be so simple. Any method you use could improve the number of games you win overall while simultaneously making it highly unlikely that you pick every single game right. Whatever knowledge you use to pick your bracket, the method could actually increase the probability of missing one or two of the wildly improbable outcomes that occur every year.

Ablowitz compares it to the stock market. “Say you look at a mutual fund, and they have these guys who are professional stock pickers. They have all the data on these companies, just like somebody might have data on basketball teams, but most mutual fund companies, active traders, don’t do as well as the averages like the S&P 500. … The average does better than the stock pickers.”

You might chalk it up to luck, the inevitable randomness of the universe in determining the outcome of March Madness. But even though no one is likely to pick a perfect bracket before the sun enlarges and engulfs the Earth in about five billion years, that shouldn’t stop you from taking that 1 in 9.2 quintillion shot at perfection.

Jay Bennett | | READ MORE

Jay Bennett was the associate web editor, science, for Smithsonian.