The NCAA Tournament presents a great real-live way to get students to think about big numbers.
In the abstract, it can be easy to confuse a trillion and a quadrillion and a quintillion — those are all impossibly big numbers for humans to keep straight.
But it’s important for students — and for everyone, really — to keep big numbers straight.
A year ago, 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 got the math wrong. I commented on the NPR site, and I think my comment helped NPR fix the error… but when I first read the article, here’s a screen clipping of what it said — it’s an example of how easy it is to not think math all the way through:
That looks good, but it has two mathematical flaws. First, it says 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).”
NPR had it right verbally, but its number was wrong — a quintillion is a 1 followed by 18 zeros, and NPR had 15 zeroes following its 147.
When we think about zeroes, it’s not that complicated: it goes million (6 zeroes), billion (9), trillion (12), quadrillion (15), quintillion (18).
NPR then made 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.”
Here’s what 9 trillion looks like: 9,000,000,000,000 — a nine followed by 12 zeroes. To go from 147 quintillion (18 zeroes) to 9 trillion (12 zeroes) 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 way to figure out the number of possible outcomes is to take the total number of games played and raise 2 to that power. One game? Two possible outcomes. Two games? Four possible outcomes.
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 in the NCAAs, 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 were 68 teams, so that’s 2 to the 67th possibilities, which is the 147 quintillion number. If there are 64 teams (which there were earlier today), 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.
As the round of 64 comes to a conclusion, nobody in the 11 million entries on ESPN still has a perfect bracket. That should not be surprising, because the odds of picking all 32 games correctly is one in “2 to the 32nd power,” which works out to 1 in 4,294,967,296 — that’s 4 billion, 294 million.
When we get to the Final Four, there will be 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 should not need math professors to explain this.
Middle school students can handle this sort of math — they 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).