Large Numbers Madness

For sports fans, it’s the most wonderful time of the year: the first week of the NCAA men’s basketball tournament!

The bracket for the 2021 NCAA men's basketball tournament
This year’s NCAA men’s basketball tournament bracket – click for a larger version

Last night, the qualifying teams were announced, along with their matchups in the first round. And here it is – click for a larger version that you can read.

That means that today, millions of people in the U.S. and around the world are carefully studying this bracket, trying to pick the winners of each game in each round, for fun and profit. And so a question that often comes up this time of year is this:

What are the chances of picking a perfect NCAA bracket?

That is, what is the probability of correctly predicting the winner of every single game, from the first first-round game through to the championship?

Seemingly simple games can produce absurdly large numbers quickly, as we saw earlier with a simple deck of cards. March Madness doesn’t produce numbers quite that absurd as the cards did, but still so absurd that it’s hard to wrap your mind around just how large the number is.

The UMBC Retrievers celebrate after winning the first 16-vs-1 matchup in NCAA men's basketball tournament history
Miracles do happen: (16) UMBC 74 (1) Virginia 54 in 2019

Let’s ignore the play-in games on Wednesday night, since those exist only to confuse and annoy. That leaves 64 teams, which means that to crown a champion it will take 63 games. Assume that in each game, each team has a 50-50 chance of winning, meaning you can flip a coin to predict the winner. (This is not a realistic assumption, since for example, the 1-seed has won 139 out of 140 games – but since miracles can happen, let’s go for it.)

That means that the probability of getting every game right is 1 in 263, which is

1 in 9,223,372,036,854,775,808

That’s one correct choice out of 9 quintillion 223 quadrillion 372 trillion 36 billion 854 million 775 thousand 808 possible combinations.

Loki from the Avengers movie series
What kind of divine being would be annoying enough… oh, right. Never mind.

How absurdly huge is this? See below for the calculations, but in short, let me put it this way.

Let’s say that some really annoying divine being decided they wanted to cheat by getting humans to pick every possible bracket. They would have to kidnap every human on Earth – all eight billion of us – and endow us all with eternal life and super-speed to pick brackets.

Then they would magically transport us back to the year 1983, just in time to watch one of the most famous moments in college basketball history:

Then we would get to work – all eight billion of us – picking one possible NCAA bracket per second. No stopping to eat or sleep, just filling out March Madness brackets once per second, all day all night all year. It could have happened earlier, but in the worst case it would be just about now – after about 38 years of work – that someone, somewhere would get a perfect bracket. And that’s just for this year; picking next year’s bracket could keep us busy until the year 2060.

But take heart – you don’t have to get them all right to win your office pool. As they say, you and your friend are being chased by a bear, you don’t have to outrun the bear. You just have to outrun your friend.

Replace the bear in that analogy with the laws of probability, and good luck with your picks this year!

The (highly approximate) calculations

  • There are 263 possible combinations, which is about 9 x 1018
  • There are 8 billion people on Earth (8 x 109)
  • So everyone picking one bracket per second is 8 x 109 brackets per second
  • There are about 30 million seconds in a year (3 x 107)
  • So every human picking one bracket per second for a year multiplies to about 2.4 x 1017 brackets
  • 9 x 1018 brackets divided by 2.4 x 1017 brackets per year is about 38 years

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