March Madness The Perfect Bracket

March Madness kicks off every Spring, and within a few hours of the start of the NCAA Basketball Tournament millions of fans across America will be despairing over their already-ruined brackets. Americans fill out 40 million March Madness brackets every year, trying to correctly predict the winners of each of the 63 games in the tournament.

Many websites offer prizes for the best bracket submission, but even the winners aren’t perfect., for example, had over 4.6 million submissions in 2009, but Joseph Taylor, the winner of the $10,000 grand prize, still got five picks wrong.

So what are the odds of filling out a perfect bracket? Well, the answer depends on how you go about the math. When Wall Street Journal writer Carl Bialik looked into the question, he got answers ranging from one in 150 million to one in 9 million trillion (that’s a “9” followed by 18 zeroes). Either way, the odds are less than great, but let’s try to nail down the answer a bit more precisely.

There are a lot of different ways you can pick a bracket. You could just flip a coin to make every pick—that would yield the million trillion estimate. You could use some sort of power rating system—Bialik calculated that your odds would improve to one in 722 billion if you did that. Or you could rely on each team’s seeding—after all, the NCAA tournament is divided into four regions, with the teams in each region seeded 1 through 16, giving us a simple way of ranking all the teams.

If we look at the historical results, we see that the seeds work very well in each round—that is, the higher seeded team wins more often than not. Here, for example, are the numbers for the first round:

SeedWin% SeedWin%
11.00 160.00
20.96 150.04
30.85 140.15
40.78 130.22
50.69 120.31
60.69 110.31
70.61 100.39
80.50 90.50

Clearly, if you wanted to maximize your chances of having a perfect bracket you would choose the higher seed in every first round matchup, except the 8/9 game where you would be indifferent. Still, even if you followed such a strategy your odds of escaping just the first round unscathed would be very small—just 1 in 13,040—and there would still be five rounds to go.

The second round looks much like the first—No. 1 seeds generally beat No. 8/9 seeds, No. 2 seeds generally beat No. 7 seeds, No. 3 seeds generally beat No. 6 seeds, and No. 4 seeds generally beat #5 seeds (though the advantage in those last two pairs is very small). Again, you’d be best off picking all the favorites in round two, but the odds they’d all win would still be only about 1 in 1,032. Overall, your odds of having a perfect bracket through two rounds are just 1 in 13,460,000.

I’ll skip the calculations for each successive round—suffice it to say, the higher seed is always the favorite. Still, if you follow the optimal strategy of always picking the higher seed to win, your odds of filing a perfect March Madness bracket are just 1 in 35,360,000,000—almost 18 times worse than your odds of being killed by a waterspout in a year (1 in 1,988,000,000).

So whenever it is that your bracket is first blemished—whether in the first game of the first round, or sometime later down the road—don’t be too disappointed. And please, watch out for those waterspouts.

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