3 Numerical Fun Facts About the NCAA Tournament
How can you quickly figure out how many games are needed to determine the winner of a tournament? What are the odds of predicting the winner of each and every game? And how does this explain why your NCAA tournament bracket was so wrong? Keep on reading to find out.
Jason Marshall, PhD
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3 Numerical Fun Facts About the NCAA Tournament
It’s that time of year again—when people all across the nation who don’t really know or care much about college basketball watch a lot of college basketball. So if you, like millions of other people, have spent a significant number of hours over the past few weeks watching the NCAA tournament, you’re in good company.
Because it’s March. Wherein there is madness. But is there math in this March Madness? Are there perhaps numerical fun facts waiting to be discovered? Indeed, there are lots of them! And today we’re going to talk about three of my favorite—including one that will help you understand why that bracket you spent so much time toiling over turned out so terribly.
Tournament Math
Before we get into the nitty-gritty details of the NCAA basketball tournament, I’d like to take a few minutes to talk about tournaments in general. In particular, I’d like to talk about how single-elimination or knockout style tournaments are set up. Because there’s some pretty cool math within their structure.
To begin with, barring some fancy finagling with byes or the like, every single-elimination tournament—whether it be the NCAA tournament, the knock-out stage of the World Cup, a major tennis championship, or whatever—starts with an even number of teams. Which makes sense because every team must have another team to play in the first round.
But not only are there an even number of teams, the number of teams must actually be a power of 2 (again, barring some more of that fancy finagling … which, we’ll soon see, the NCAA tournament actually contains a bit of). If the tournament takes, say, 4 rounds to determine a winner, we know that there must be 24=16 competitors—since 16 teams in the 1st round beget 8 teams in the 2nd (quarter-final) round which beget 4 teams in the 3rd (semi-final) round which beget 2 teams in the 4th (and final) round determining the winner.
How to Quickly Calculate the Games in a Tournament
This leads us to the question that will give rise to our first numerical fun fact for today: How many games are required to determine the NCAA tournament winner? There are (at least) two ways to think about this: the hard way and the easy way.
Let’s start with the hard way. As we’ve seen, once we know the number of teams in a tournament, we can fairly simply figure out the number of rounds it’ll take to determine the winner. Namely, since the number of teams in a tournament must be equal to 2 raised to the power of the number of rounds played, we know that, for example, a 64 team tournament like the NCAA tournament must have 6 rounds—since 26=64 teams. (I know, technically the NCAA tournament starts with 68 teams, but 4 of the low-seeded teams are quickly whittled down in a sort of pre-tournament tournament … leaving us with 64 teams in the main bracket.)
In any 64 team tournament there must be 1 winning team … and therefore 63 losing teams.
Now, if you think about it, you’ll see that each round must contain half as many games as there are teams in that round. So there must be 8 games amongst the 16 teams in the NCAA tournament’s “Sweet 16,” 4 games amongst the 8 teams in the tournament’s “Great 8,” and so on. Putting this all together, we see that a 6 round tournament must have 32+16+8+4+2+1=63 games in total.
That’s the hard—or at least somewhat laborious—way to figure out the answer. The easy way is to realize that in any 64 team tournament there must be 1 winning team … and therefore 63 losing teams. How do you get 63 losing teams? You have to play 63 games!
What Are the Odds of Predicting a Perfect Bracket?
If you, like countless other people spent time this spring filling out an NCAA tournament bracket to predict the various winners and losers of these 63 main bracket games, you may have come to the realization that this is a lot of games to predict—and that having to predict so many games means that filling out a perfect bracket is a really, really (really) hard thing to do!
How hard exactly? Well, let’s think about the math. Let’s assume that you have a 50/50 shot at “guessing” the winner of each game. In truth, if you study up on the teams you can do a bit better than 50/50 with some of the match-ups, but let’s keep things simple. In this case, the probability of your predicting the winner of the first game correctly is 1/2. After two games, there are four possibilities: (1) Right on both games; (2) Right on the first but wrong on the second; (3) Wrong on the first but right on the second; and (4) Wrong on both. So after only 2 games, your predictions will be imperfect 75% of the time.
The general trend is that with each additional game played, the number of possible brackets grows by two. So there are 2 possible brackets in a 1 game tournament, 22=4 possible brackets in a 2 game tournament, 23=8 possible brackets in a 3 game tournament, and so on. The number of possible brackets is always equal to 2(Number of Games). Which brings us to our second numerical fun fact of the day: There are 263 or a bit more than 9 quintillion (which is 9 billion billion) possible brackets in a 63 game tournament!
Why Was Your NCAA Tournament Bracket So Wrong?
So as I mentioned before, it’s really hard to perfectly predict an NCAA tournament bracket. As far as I can tell after digging around the Internet for a while, it has actually never been done … at least as far as anybody seems to know about. And that really shouldn’t be a surprise. In fact, it would be amazing if it had been done before.
The odds of predicting a perfect NCAA tournament bracket are roughly the same as the odds of winning 63 coin tosses in a row.
After all, 9 quintillion is a HUGE number. And your odds of predicting a perfect NCAA tournament bracket are about 1 in 9 quintillion. How long are those odds? Think about it this way: Imagine you correctly predict 10 coin tosses in a row. This is definitely possible to do, but it doesn’t seem very likely, right? Now keep tossing the coin … another 10 times … and then another … and another … and another … and yet 10 more times … and correctly predicting the outcome each time without a single miss. And then for good measure do it 3 more times. That’s a lot of perfectly predicted coin tosses.
Which brings us to our final numerical fun fact for the day: The odds of predicting a perfect NCAA tournament bracket are roughly the same as the odds of winning 63 coin tosses in a row. Those are indeed really, really (really) long odds—which is why nobody has ever managed to predict a perfect bracket. And it more than likely explains why your and everybody else’s brackets turned out so wrong!
Wrap Up
OK, that’s all the math we have time for today.
For more fun with math, please check out my book, The Math Dude’s Quick and Dirty Guide to Algebra. Also, remember to become a fan of The Math Dude on Facebook and to follow me on Twitter.
Until next time, this is Jason Marshall with The Math Dude’s Quick and Dirty Tips to Make Math Easier. Thanks for reading, math fans!
Basketball image from Shutterstock.
About the Author
Jason Marshall is the author of The Math Dude’s Quick and Dirty Guide to Algebra. He provides clear explanations of math terms and principles, and his simple tricks for solving basic algebra problems will have even the most math-phobic person looking forward to working out whatever math problem comes their way.