In the midst of March Madness, just as the NCAA tournament was about to begin, I read a piece on NPR about the odds of choosing a perfect NCAA bracket.
It’s a fine piece, except for one thing — it gets the math wrong. I commented on the NPR site, and I think my comment helped NPR get the math right… but when I read the article, here’s a screen clipping of what it said:
Here’s what I wrote in my comment:
This is a fun story, but it’s not mathematically accurate. You note that the odds of choosing the entire 68-team field is “about 147 quintillion to one (that’s 147,000,000,000,000,000:1).”
That’s right verbally, but your number is wrong — a quintillion is a 1 followed by 18 zeros (you have 15 zeroes following your 147). It goes million (6 zeroes), billion (9), trillion (12), quadrillion (15), quintillion (18).
Your story then makes the mistake of saying that “The odds get slightly better (about 9 trillion to one) if you ignore the play-in games and just look at the field of 64.”
9 trillion is 9,000,000,000,000 — that’s a nine followed by 12 zeroes. To go from 147 quintillion (18 zeroes) to 9 trillion would be more than “slightly” improving your odds. But in fact, that’s not what’s happening — when you ignore the play-in games, your odds go from 147 quintillion to 9 quintillion. They improve by a factor of 16. And the reason is that there are four fewer games played.
The algorithm for the number of possible outcomes is to take the total number of games played and raise 2 to that power. The number of games played is the number of teams minus one, because once a team loses it goes home; if there are 64 teams playing, there will be 63 teams that end their season with a loss and one national champion. That makes 63 games.
If you count the play-in games, there are 68 teams, so that’s 2 to the 67th games, which is the 147 quintillion number. If there are 64 teams (which there will be after Wednesday night’s games), you raise 2 to the 63rd power to get the total number of possible outcomes — and that’s 9,223,372,036,854,775,808. So the odds of filling out a perfect bracket are a bit worse than 1 in 9.2 quintillion.
When we get to the Final Four, there are four teams left, so we’d raise 2 to the third power and get 8 possible outcomes. When we get to the Sweet 16, there are 15 games to be played, and 2 to the 15th power is 32,768.
Here’s a 3-minute video from math professor Jeff Bergen at DePaul explaining this same idea, but we really shouldn’t need math professors to explain this. I’m opening a new middle school in a few months in Durham, NC (called Triangle Learning Community), and I’m confident that my students can handle this sort of math.
When it comes to big numbers, we can keep things straight — we just need to slow down and think things through.
POSTSCRIPT: Below is a screen shot showing how NPR fixed its mistakes, possibly in response to my comment on the NPR site. I feel like a good digital citizen 🙂
Here’s a look at the closest anyone has come to getting a perfect bracket (55 out of 63 games right).