Bracket IQ

BACK TO Bracket IQ
basketball-men-d1 flag

Daniel Wilco | NCAA.com | March 25, 2019

A perfect NCAA bracket: The absurd odds of the March Madness dream

Exploring the absurd odds of a perfect bracket

Every year, millions of people fill out a bracket for the NCAA tournament. If you're like us, you hear that little voice saying, “What if I became the first person ever to fill out a perfect bracket? This could be the year!”

That little voice knows one thing: No one has gotten a verifiably perfect bracket in the history of the NCAA tournament. But it also has one thing very wrong: This will not be the year. And neither will next year, or any in the next millennium.

RELATED: 1 NCAA tournament bracket stays perfect into Sweet 16 for first time ever. Here’s how it happened

Yes, it is technically possible, and even absurdly overwhelming odds don’t mean it couldn’t theoretically happen this year. But we’re pretty confident in saying that it won’t.

How crazy small is the chance?

It’s pretty hard to calculate the exact odds of filling out a perfect bracket. Your chances will increase with more knowledge of the current teams, the tournament’s history, and an understanding of the sport itself. For instance, before UMBC’s historic upset of Virginia last year, it was practically a guarantee that all four 1 seeds would win their matchups (they’re still 135 for 136 through the modern tournament’s history), giving you four automatically correct games to start off with. But that type of knowledge is near impossible to quantify or accurately factor into an equation.

We'll get to advanced calculations that attempt to take knowledge into account later on, but to get a better understanding, let’s first look at the most basic calculation.

MORE: This is the longest we believe a March Madness bracket has ever stayed perfect

What are your odds if you had a perfect 50-50 chance of guessing every game correctly? Well that would depend on the number of total possible bracket permutations for the tournament.

So how do we calculate this? We'll look at a small sample bracket first. Like the NCAA tournament, our sample bracket will be a single-elimination tournament, but it will feature just four teams.

Let's fill out all of the possible outcomes for that tournament's bracket:

All permutations for a 4-team, single-elimination bracket

That gives us eight bracket permutations.

For a small field of just four, that's easy to sketch out. But even if we just double the field to eight teams, the results are daunting.

With eight teams, we go from eight bracket permutations to 128:

All 128 permutations for an 8-team bracket

That's the fun thing about exponents: they increase exponentially.

(And for those of you who are so bored you wanted to zoom in on each of those 128 brackets, no we didn't take the time to actually fill out each one correctly. That would take too long. That's kinda the point here.)

But instead of just sketching out every possible outcome of every game, we can also get the number of possible brackets using those exponents.

All we have to do is take the number of outcomes for a game (2) and raise it to the power of the number of games in the tournament. For our first example, that's 2^3, which gives us 8. For the second, it's 2^7, giving us 128.

Now let's apply that to the modern NCAA tournament.

Since 2011, the NCAA tournament has had 68 teams competing in its field. Eight of those teams compete in the “First Four” — four games that take place before the first round of the tournament. Virtually all bracket pools disregard these games and only have players pick from the first round, when 64 teams remain.

Therefore, there are 63 games in a normal NCAA tournament bracket. 

2019-march-madness-bracket

As such, the number of possible outcomes for a bracket is 2^63, or 9,223,372,036,854,775,808. That’s 9.2 quintillion. In case you were wondering, one quintillion is one billion billions.

If we treated the odds for each game as a coin flip, that makes the odds of picking all 63 games correctly 1 in 9.2 quintillion. Again, this is not a completely accurate representation of the odds, as any knowledge of the sport or tournament’s history improves your chances of picking games. But it is one of the easiest to quantify, so let’s have some fun with it.

How crazy are 1 in 9.2 quintillion odds?

Let's do another visual experiment. 

Here is a picture of one dot:

One pixel

Missed it? No worries, we'll help you out. It's inside the circle.

One pixel circled

Okay, now let's take a look at one million of those dots:

One million pixels

Definitely easier to see.

But we still have a long way to go. Now imagine a new picture where each one of those dots in the picture above contained one million dots itself. One million million dots. Also known as a trillion.

We'd need 9.2 million of those new pictures to get 9.2 quintillion dots.

Not impressed yet? Fine.

A group of researchers at the University of Hawaii estimated that there are 7.5 quintillion grains of sand on Earth. If we were to pick one of those at random, and then give you one chance to guess which of the 7.5 quintillion grains of sand on the entire planet we had chosen, your odds of getting it correct would be 23 percent better than picking a perfect bracket by coin flip.

These numbers are way too large to fully wrap your head around, but here are a handful of other statistics for reference, compared to 9.2 quintillion.

  • There are 31.6 million seconds in a year, so 9.2 quintillion seconds is a quick 292 billion years.
  • There have been 5 trillion days since the Big Bang, so repeat the entire history of our universe 1.8 million times.
  • The Earth’s circumference is approximately 1.58 billion inches, so you’d have to walk around the planet 5.8 billion times.
  • As of 2015, the best estimates for the number of trees on the planet was three trillion. Imagine that there was one single acorn hidden in one of those three trillion trees, and you were tasked with finding it on the first guess. Your odds of success are approximately three million times greater than picking a perfect bracket.

But we’ve already said that the 1 in 9.2 quintillion figure is a bit disingenuous. Others have tried to refine the rough estimate.

Georgia Tech professor Joel Sokol (that's him above) has worked for years on a statistical model to predict college basketball games, and he says that the best models we have today are only right three quarters of the time, at best.

"In general, about 75 percent is where you’ll get for essentially any model," Sokol said. "Any of the best ones. Which is partly what makes people think that about a quarter of tournament games are upsets. It might be a little higher or a little lower, but give or take, it’s close to 75 percent, where the best models can pick out which teams are better than others and then it’s just a question of whether the ball bounces the right way, who is playing better that day, whatever, whether you get the upset that day or not.”

Sokol said that using a model that predicts regular-season games correctly 75 percent of the time would give you odds of getting a perfect bracket anywhere between 1 in 10 billion to 1 in 40 billion. Much, much better than 1 in 9.2 quintillion, but still crazy high. So high that Sokol doesn't believe it will ever happen.

"Even the most optimistic number I’ve seen, which is about 1 in 2 billion, that means give or take, if you want a 50-50 chance of ever seeing it in your life, you have to go through 1 billion NCAA tournaments," he said. "And you might say, well there’s millions of people filling these brackets out every year, but really there’s not that much variation in the brackets, compared to how many there could be."

About that, last year, of the millions of brackets entered into our Bracket Challenge Game, 94.4 percent were unique. Even with 94.4 percent of millions of brackets being unique, we only covered 0.0000000000182 percent of all possible bracket permutations. So close.

Speaking of Bracket Challenge Game users, we can use that data to get another estimate on the odds of a perfect bracket. We have the pick history for millions of players over the past five years.

We looked at the average user’s pick accuracy for all 32 first round games over the past five years (that’s 160 games per user). Then we weighted those percentages by the frequency of that matchup’s seed differential. For example, a 5 vs. 12 game has a seed differential of 7. There have been 222 games with a seed differential of 7 in the NCAA tournament’s modern history.

Then, we combined all the percentages to give us the average player’s accuracy for an average game: 66.7 percent. Not bad. Now, for the odds of a perfect bracket using that percentage:

667^63 = 0.00000000000831625.

That's equal to odds of 1 in 120.2 billion —  70 million times better than if every game was a coin flip.

How attainable are odds of 1 in 120.2 billion?

If every person in the United States filled out a completely unique bracket that was 66.7 percent accurate, we'd expect to see a perfect bracket 366 years from now. You know, if March Madness is still happening in the year 2385.

But until all Americans come together to proficiently fill out unique brackets, keep ignoring that little voice in your head, and take solace in the fact that you don’t have to be anywhere near perfect to win. In the past eight years of our Bracket Challenge Game, winners have averaged just 49.8 correct games in their brackets. Now that is achievable.