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The Odds Of A Perfect Bracket Are Too Infinitesimal For My Puny Primate Brain

Perfect brackets: They’re rather hard. You know this. Before we get into the whole “you have a one-in-who-gives-a-craptillion chance of winning” part of the story, though, let’s talk about why it’s so hard to grasp big numbers like this.

Generally speaking, without a way to anchor a number to an everyday concept, people tend to have a hard time with Very Big Numbers. For some of them, like my favorite thing of all things the Powerball lottery, it’s still doable. For instance, your odds of winning the Powerball lottery are roughly equivalent to picking a random adult who lives in the U.S. and Canada and that person being you. It’s the probability of selecting a random person on earth, and that person having been in your freshman biology class. Powerball, with its 1 in 292 million probability of success, approaches the upper limit of what we can convey.

My personal limit is 1 in 7 billion, because that is how many people are on earth — essentially the biggest number I feel I have any hope of grasping. Any more than that and you’ve got to do one of those “line up five decks of shuffled cards” monstrosities, and by then you’re just grasping at straws — we get it, dude, it’s unlikely.

Last year I got to interview Randall Munroe — a guy who regularly confronts this kind of problem — and he talked about it better than anyone I can think of:

One thing that bothers me is large numbers presented without context. We’re always seeing things like “This canal project will require 1.15 million tons of concrete.” It’s presented as if it should mean something to us, as if numbers are inherently informative. So we feel like if we don’t understand it, it’s our fault.

But I have only a vague idea of what one ton of concrete looks like. I have no idea what to think of a million tons. Is that a lot? It’s clearly supposed to sound like a lot, because it has the word “million” in it.

You should read that whole interview, because he’s got a brilliant take on a problem we deal with more often than we’d prefer to admit, but his key point is this: “A good rule of thumb might be, ‘If I added a zero to this number, would the sentence containing it mean something different to me?’ If the answer is ‘no,’ maybe the number has no business being in the sentence in the first place.”

That being said, here is a series of sentences about the number 9.2 quintillion, and how you might work your head around that many possible brackets.

We should start small, then work our way up. Let’s take the most basic entry point, a single round-of-32 game. There are two possible ways to fill in that part of the bracket.

hickey-perfectbracket-1a

Next, we go to one-fourth of a division: four teams vying for one spot in the Sweet 16 over three games. There are eight possible configurations of this bracket — two possible ways to fill in the first game, two ways to fill in the second, and then two possible ways to fill in the final game, given that you only have two choices from the first four teams. That’s eight configurations, or 2^3. Your probability of picking one of these perfectly is thus 1 in 8. Given that you filled out 16 of them, check your bracket. You probably got at least a few perfect!

hickey-perfectbracket-1b

Next, we have a half of a region. We have two of those previous bracket chunks — each with eight possible configurations — and two ways to pick who goes on to the Elite Eight. This means we have 8 times 8 times 2 possible configurations, or 128 possible ways to fill out this section. Just another quick note: 128 is 2^7, seven being the number of games in this chunk of bracket. Maybe someone in your office got one of these perfectly! But it’s starting to get pretty hard, right?

hickey-perfectbracket-1c

Now we’re getting to an increasingly hard part: calling a region perfectly. One region has two of those smaller chunks — with 128 combinations each — plus two choices in the last game. This gives us 128 times 128 times 2 possible configurations, or 32,768 ways to fill out a division. That’s 2^15, which, you guessed it, is the number of games in this chunk. You filled out eight of these. If everyone in a company of 5,600 people, which is a little smaller than ESPN, filled out a bracket randomly, there’s a 50-50 shot that at least one of them gets one division perfect.

hickey-perfectbracket-1d

Moving up to half a bracket: With 31 games to call, there are 2^31 possible combinations — 2,147,483,648, to be exact. That means that randomly guessing winners, you are about one-seventh as likely to get half a bracket right as you are to win the Powerball jackpot (1 in 292 million, you’ll recall). It is half as likely as randomly selecting a resident of the Americas and having that person be Bill Murray.

hickey-perfectbracket-1e

This brings us to the final bracket. With 63 games and 2 possible selections for each one, that’s 9,223,372,036,854,775,808 possible combinations. Your bracket is one of these, and the perfect bracket is one of these, but it is highly unlikely that they are the same bracket, is what I’m saying.

hickey-perfectbracket-1f

Here is some perspective on that figure. (N.B. Probably this will not help at all.)

Still, while 9.2 quintillion is the number that gets thrown around, it sucks because it’s so huge and also, honestly, somewhat pointless. There’s a reason people mostly perform better than the expected rate of bracket busting: They aren’t selecting randomly. There are different probabilities for each game, and people are aware it’s not a coin flip between first-seed Kansas and 16th-seed Austin Peay.

Estimates for the legitimate probability of a perfect bracket vary. Based on reporting from USA Today, Duke math professor Jonathan Mattingly puts it at 1 in 2.4 trillion — I love this estimate, and you’ll see why in just a second — while a professor at DePaul University puts it as low as 1 in 128 billion.

I have no idea what it actually is, but using the pre-tournament probabilities of each team in the FiveThirtyEight interactive, I can tell you the probability of a perfect chalk bracket. That is, if in every game you selected the team that FiveThirtyEight gave the higher pre-tournament likelihood of winning, what’s your actual probability of getting a perfect bracket? Based on the assigned probabilities of advancement at each game, picking the likeliest team to advance and multiplying through all the pre-tourney probabilities, the odds that every one of our predictions would be right was 1 in 2,460,838,227,877 (2.5 trillion). This is delightful because it’s right in the ballpark of Mattingly’s estimate. Indeed, it’s basically right on the pitcher’s mound with it.

So, the gist: Numbers are hard; big numbers are pointless and not attuned to the human brain; this should not make you feel bad; you should be proud if you nailed half of a region and thrilled if you got a whole one; and don’t hold your breath for anything more. See you next year or maybe earlier, if the Powerball gets big enough.

Walt Hickey was FiveThirtyEight’s chief culture writer.

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