The absurd odds of a perfect NCAA bracket

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

This little voice knows one thing: no one has had a provably perfect bracket in NCAA tournament history. But it also has a major flaw: this will not be the year. And neither in the next year nor in the next millennium.

BRACKETS: Print out the official March Madness bracket

Yes, it’s technically possible, and even absurdly overwhelming odds don’t mean it couldn’t theoretically happen this year. But we’re pretty confident to say that won’t be the case.

How crazy small is the chance?

Here is the TL/DR version of the odds for a perfect NCAA bracket:

  • 1 in 9,223,372,036,854,775,808 (if you’re just guessing or flipping a coin)
  • 1 in 120.2 billion (if you know anything about basketball)

Your odds increase with more knowledge of the current squads, the history of the tournament, and an understanding of the sport itself. For example, prior to UMBC’s historic uproar against Virginia last year, all four 1-seeded players were virtually guaranteed to get their matchups would win (they still stand 135 to 136 in modern tournament history), giving you four automatically correct games to start with. But this kind of knowledge is nearly impossible to quantify or accurately fit into an equation.

We’ll get to more advanced calculations later that try to account for knowledge, but to get a better understanding let’s take a look at the most basic calculation first.

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

So how do we calculate that? First, let’s look at a small sample mount. Like the NCAA tournament, our example bracket will be a single-elimination tournament, but will only feature four teams.

Let’s fill in all possible results for this tournament’s bracket:

That gives us eight bracket permutations.

For a small field of only four, this is easy to sketch. But even if we just double the field to eight teams, the results are scary.

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

That’s the beauty of exponents: they grow exponentially.

(And for those of you so bored that you wanted to enlarge each of those 128 brackets, no, we didn’t take the time to fill each one in correctly. That would take too long. That’s kind of the point here. )

But instead of just sketching every possible outcome of each game, we can also use these exponents to get the number of possible brackets.

All we have to do is raise the number of outcomes for a game (2) 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.

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

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

Since 2011, 68 teams in his area have competed in the NCAA tournament. Eight of these teams compete in the “First Four” – four games that take place before the first round of the tournament. Virtually all bracket pools ignore these games and only let players pick from the first round if 64 teams remain.

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

Thus, the number of possible outcomes for a parenthesis is 2^63, or 9,223,372,036,854,775,808. That’s 9.2 quintillion. In case you are wondering, a trillion is a billion billion.

If we treat the odds for each game like a coin toss, the chance of correctly picking all 63 games is 1 in 9.2 trillion. Again, this is not an absolutely accurate representation of the odds as any knowledge of the sport or tournament history will improve your chances of selecting matches. But it’s one of the easiest to quantify, so let’s have some fun with it.

How crazy are the odds of 1 in 9.2 quintillion?

Let’s do another visual experiment.

Here is a picture of one point:

missed it? Don’t worry, we’ll help you further. It’s inside the circle.

Okay, now let’s take a look at a million of these points:

Definitely easier to see.

But we still have a long way to go. Now imagine a new image where each of these points in the above image itself contains a million points. A million million points. Also called a trillion.

It would take 9.2 million of these new images to get 9.2 quintillion points.

Not impressed yet? Fine.

A group of researchers from the University of Hawaii estimated that there are 7.5 trillion grains of sand on Earth. If we randomly picked one of these and then gave you a chance to guess which of the 7.5 quintillion grains of sand on the entire planet we picked, your odds of getting it right would be 23 percent better than picking a perfect bracket by tossing a coin.

Those numbers are far too big to fully capture, but here are a handful of other stats for reference, compared to 9.2 quintillion.

  • There are 31.6 million seconds in a year, so 9.2 quintillion seconds is almost 292 billion years.
  • 5 trillion days have passed since the Big Bang, so repeat the entire history of our universe 1.8 million times.
  • The circumference of the earth is approximately 1.58 billion inches, so you would have to go around the planet 5.8 billion times.
  • In 2015, the best estimate for the number of trees on the planet was three trillion. Imagine there was a single acorn hidden in one of those three trillion trees, and you were tasked with finding it first time. Your odds of success are approximately three million times greater than choosing the perfect mount.

But we have already said that the number 1 in 9.2 trillion is a bit disingenuous. Others have attempted to refine the rough estimate.

Georgia Tech professor Joel Sokol (that’s him above) has spent years working on a statistical model to predict college basketball games, and he says the best models we have today are, at best, correct three-fourths of the time.

“Generally, it’s about 75 percent that you get for pretty much every model,” Sokol said. “One of the best. Which partly leads people to believe that about a quarter of tournament games are upset. It could be a little bit higher or a little bit lower, but give or take, it’s almost 75 percent where the best models are able to pick out which teams are better than others and then it’s just a matter of whether the ball bounces in the right direction, who’s playing better that day, whatever, whether you’re upset about that day or not.

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

“Even the most optimistic number I’ve seen, which is about 1 in 2 billion, that’s give or take, if you want a 50-50 chance of ever seeing it in your life, you have to do 1 billion NCAA tournaments.” he said. “And you could say, well, there are millions of people who fill out these brackets every year, but there really aren’t that many differences in the brackets compared to the number that there might be.”

Over the past year, 94.4 percent of the millions of brackets that participated in our bracket challenge game were unique. Although 94.4 percent of millions of brackets are unique, we only covered 0.0000000000182 percent of all possible bracket permutations. So close.

Speaking of bracket challenge game users, we can use this data to get another estimate of the odds of a perfect bracket. We have pick history for millions of players over the last five years.

We looked at the user’s average 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 5v12 game has a seed differential of 7. In modern NCAA tournament history, there have been 222 games with a seed differential of 7.

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 this percentage:

667^63 = 0.00000000000831625.

That equates to odds of 1 in 120.2 billion – 70 million times better than if every game were a coin toss.

How realistic are odds of 1 in 120.2 billion?

If every person in the United States filled out a completely unique bracket with an accuracy of 66.7 percent, we would expect to see a perfect bracket in 366 years. Do you know if March Madness is still happening in 2385?

But until all Americans come together to competently fill in unique brackets, ignore that little voice in your head and console yourself with the fact that you don’t have to be anywhere near perfect to win. Over the past eight years of our Bracket Challenge play, winners have averaged just 49.8 correct games in their brackets. Now that is achievable.

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