The Odds of a Perfect Bracket are Higher Than You Think
If you treat every game as a coin flip, the odds of a perfect bracket in the NCAA March Madness tournament are roughly in 1 in 9 quintillion. That’s the same thing as pulling off three “one in a million” things in a row. And then flipping a coin and getting heads three times in a row.
But treating every game as a coin flip is not an accurate representation of reality. The 1 seeds are almost guaranteed to beat the 16 seeds in the first round; that game is more a gimme than a coin flip. Because not every game is a tossup, the odds of a perfect bracket are higher than you think.
How much better? Our analysis shows that the odds are closer to 1 in 28.5 trillion instead of 1 in 9 quintillion. Though, this is still an extreme longshot, this means that the true odds of a perfect bracket are nearly 315,000 times more likely than previously thought. The graphic below shows how the odds of a perfect bracket decay by round. The data in this figure show “the probability of having a perfect brack through round X” where X is shown on the horizontal axis.
Even being perfect through round 1 is a “1 in a million” feat.
But returning to the topic of this article, why are the odds of a perfect bracket so much higher than previously thought? The next section gives an intuitive approach to understand why this is the case. After that we show the mathematics of how we calculated our results.
Intuition For Perfect Brackets Being More Likely
In the beginning, we described broadly the reason that the odds of a perfect bracket are much higher than most people think. Almost every website or article or talking head that talks about the odds of a perfect bracket assumes that every game is 50-50.
The way that the “1 in 9 quintillion” odds of a perfect bracket is computed is by computing the odds of predicting 63 straight coin flips (63 is the number of games in the NCAA tournament). The probability of predicting 1 coin flip right is 0.5. Predicting 63 correct in a row is 0.563.
A Hypothetical Scenario
To show why this line of thinking is wrong, consider what would happen if there was one absolutely dominant team in the field who was nearly guaranteed to win every game. What would happen?
Everyone would pick this team to win the tournament. More specifically, the 6 games in which the dominant team plays are free wins. This means that to pick a perfect bracket, you only need to pick the other 57 games correctly. That means that we would only need to predict 57 coin flips in a row correctly. Doing this is 64 times more likely than predicting 63 coin flips in a row!
A Second Hypothetical Scenario
Of course there is never a team that is guaranteed to win every game, so the above scenario might feel a bit contrived. What happens in reality, though, is that most matchups have a favorite and an underdog. The favorite is more likely to win than the underdog. Even if the probability of the favorite winning is not 100%, the fact that there is a favorite makes it easier to correctly predict the outcome of the game.
Our second hypothetical relies on this idea. What if in every game we pick the favorite to win? And what if the favorite has a 60% chance to win each game? Then, the probability of a perfect bracket is 0.663 instead of 0.563.This might feel like a small difference, but it adds up in a huge way!
In this new hypothetical where we pick the favorite who wins 60% of the time, the odds of a perfect bracket are nearly 100,000 times better than if we treat every game as a tossup. The perfect bracket probability jumps from 1×10-19 all the way up to about 1×10-14.
Snapping Back to Reality
Neither of these hypotheticals perfectly describe reality. Sometimes (like in a #1 v. #16 matchup), the favorite will win 95+% of the time. Other times (like in a #8 v. #9 matchup) the game might truly be closer to 50-50. The odds are different in every game.
The point of the previous two hypotheticals was to convince you that perfect brackets are more likely than most people think. The point of the next section is to actually compute the odds of a perfect bracket.
Exact Math Behind the Odds of a Perfect Bracket
The difficult part about computing the odds of a perfect bracket is that the probabilities in the later rounds depend on the outcomes in the earlier rounds. That means that we can’t really predict the odds of a perfect second round without knowing the outcome of the first round. Some second rounds will be much easier to pick depending on the outcome of the first round. Some second rounds will be harder. The same is true for rounds 3 and 4 and so on.
To get around this, we use a tool which I personally think is the single most useful tool for any analyst in a STEM field to know how to use: Monte Carlo Simulation. Monte Carlo methods rely on simulating random events instead of calculating exact probabilities.
A classic application of Monte Carlo simulation is in dice games (and other games of chance, hence Monte Carlo). Suppose we want to compute the probability of rolling two dice whose sum equals 7 or 11. We could write out all the possibilities by hand. Or, we could roll the two dice 100 times and count how many times their total equals 7 or 11. The more simulations we do, the more accurate our estimate gets.
That is the entirety of the idea behind Monte Carlo simulation. Instead of computing things explicitly, set up a simulation and let the results of that simulation tell you what is going on. The key is to set up an accurate simulation. In the next section we describe how we set up our model for the NCAA tournament win probabilities.
Setting up a “Generic” Tournament
In order to simulate brackets, we need to estimate the probability that team A beats team B for any pair of teams A and B. To do this, we used historical ratings of team quality to estimate how good the top 64 teams are. In particular, the #1 team is typically 20 points better than the 64th best team.
By extrapolating this rating difference, we can infer that 1 seeds are about 1.25 points better than 2 seeds, that 2 seeds are about 1.25 points better than 3 seeds, and so on. Then, by using a conversion from “how many points better” to “win probability” we can estimate the probability that team A beats team B in a random tournament game.
Using these rules, we can simulate a “generic” NCAA March madness tournament. Doing this we get simulations of brackets which mirror reality. 12s regularly beat 5s. Almost every year a 3 or a 4 loses. There are often double digit seeds that make the sweet 16 or elite 8.
The next section describes how to use the simulation framework to estimate the odds of a perfect bracket. The next section is where things get quite mathematical.
Using our Monte Carlo Framework
This part is actually pretty straightforward, though requires just a touch of creativity. We’re going to use the law of total probability at length here. To read more, check out our previous writeup about this same topic. The key idea is to break down the odds of a perfect bracket into more bitesize chunks. For an individual bracket, the probability that it is perfect is (a) the probability that it was the bracket you picked multiplied by (b) the probability that it was the correct bracket.
To go any further we have to assume one more thing. We must assume that people pick upsets in a bracket roughly according to the probability that the upset happens in real life. That means that if a team has a 20% chance of winning, then in 20% of brackets somebody will pick this upset. Under this (fairly reasonable) assumption, the two probabilities in the previous bracket are identical.
Therefore, let pk be the probability that bracket k is the correct bracket. Then, pk is also the probability that somebody submits bracket k as their bracket. Then, the probability of a perfect bracket is . If you look closely at this formula, you’ll notice that this is equal to the average probability of any individual bracket being selected as the winner.
To use our Monte carlo simulation framework, we simply need to simulate brackets at random and keep track of the probability that the randomly simulated bracket occurs in real life. Then, over many iterations of the Monte Carlo simulation, we can estimate the probability that a random bracket is the correct one.
Results
We teased the end result above, but using the methods described here we’ve estimated that the odds of a perfect bracket are 1 in 28.5 trillion. This is roughly a probability of 3.5×10-14. Don’t get me wrong, this is still an incredible longshot. However, when compared to the previously stated odds of 1 in 9 quintillion, this is quite a bit better. In fact, it is roughly 315,000 times more likely than previously thought.
The chart below shows the probability of being perfect in a given round given that your bracket was perfect up to that point. The exact numbers are contained in a table afterwards. Also in this table is the same probabilities but assuming that every game is 50-50 (the naive approach we debunked here).
Round | Prob of Perfect Round (new model) | Prob of Perfect (50-50 model) |
|---|---|---|
1 | .00031% | 2.3×10-8 % |
2 | 0.03% | .001% |
3 | 0.98% | 0.39% |
4 | 8.5% | 6.25% |
5 | 28.0% | 25% |
6 | 52.0% | 50% |
Notice first that by far the hardest round is round 1. After this, the rounds get progressively easier to predict perfectly. Also notice that the largest difference between the two models is in the first round.
With our new model, the probability of a perfect first round is over 13,000 times more likely than the probability of a perfect round assuming every game is 50-50. This makes sense because the first round has the most lopsided matchups (16 v. 1 and 15 v. 2 for example). As the rounds go on, the models converge as the matchups tend to get more and more fair over time.
Conclusions
This article presented a new perspective on the odds of a perfect bracket. To do so, we assumed that people pick winners in their bracket roughly in proportion to how often that team. That means teams that should win 20% of the time in real life will also be picked to win about 20% of the time in brackets.
In reality, people tend to be a bit more conservative in their brackets, often preferring to pick favorites more often. This fact could lead to a different answer because the numbers used in our model differ.
To tell whether this is true or not, a more accurate approach would be to mine the data to understand how people tend to pick their brackets. With this updated model, a more accurate answer could be generated. For now, though, this model is reasonably accurate and is a pretty good guess about the real odds of a perfect bracket.
