The Probability Paradox: Why the NBA Finals (Almost) Always Go to Six Games
The NBA Finals between the Boston Celtics and Dallas Mavericks will last six games. Probably.
Cast your mind back to the 2008 NBA Finals, the last time the Boston Celtics won the title. Facing the Los Angeles Lakers, it was a series steeped in history, a renewal of the rivalry that had defined basketball for generations. The Celtics, led by the “Big Three” of Paul Pierce, Kevin Garnett, and Ray Allen, took on a Lakers team featuring Kobe Bryant and Pau Gasol.
The series was a back-and-forth affair, with neither team able to gain a clear upper hand. The Celtics drew first blood, winning Game 1 in a hard-fought battle. The Lakers responded with a victory in Game 2, tying the series as it shifted to Los Angeles. The Celtics then won two straight, putting them on the brink of a championship with a 3–1 lead.
But the Lakers, as befitting a team with their pedigree, refused to go quietly. They won Game 5 on their home court, sending the series back to Boston. In Game 6, however, the Celtics proved too strong, riding a wave of energy from their home crowd to a 131–92 victory and their 17th NBA championship.
This six-game series, filled with drama, tension, and moments of brilliance, was a testament to the unpredictable nature of sports. But what if I told you that the length of this series was not just a result of athletic competition, but a statistical probability? That the NBA Finals, regardless of the teams involved, have a surprising tendency to last for six games?
The answer lies in the captivating world of probability and combinatorics, where the unscripted drama of the hardwood meets the immutable laws of mathematics. As the Boston Celtics and the Dallas Mavericks prepare to face off in the 2024 NBA Finals, let’s explore this fascinating connection and uncover the reason behind the frequency of the six-game series.
The answer, surprisingly, is not as complicated as one might think. This is not a prediction based on the strengths and weaknesses of the teams involved, but rather a statistical inevitability rooted in the elegant world of probability and combinatorics.
To understand why, let’s step into a simplified model of the basketball universe. We’ll assume, for the sake of argument, that both the Celtics and the Mavericks have an equal 50% chance of winning any given game (i.e., a coin toss), and we’ll momentarily ignore the impact of home court advantage.
In this hypothetical world, the probability of a 4-game sweep is a mere 12.5%. This is because each team has a 6.25% chance of winning four straight games, calculated by multiplying their 50% chance of victory in each of four individual games. This is the same as getting four heads in a row — or four tails in a row — when flipping a quarter.
While a sweep is certainly possible, it’s far from the most likely outcome.
But what about a 5-game series? While sweeps are rare, five-game series are slightly more common.
This is where the beauty of combinatorics comes into play. To calculate the probability of a series ending in five games, we need to consider the number of ways in which four wins and one loss can be arranged, while also ensuring that the winning team emerges victorious in the final game. There are four such paths, each with a probability of 3.125%. Doubling this to account for the possibility of either team winning, we arrive at a 25% chance of a 5-game series.
Now, here’s where things get interesting. Using similar logic, we find that the probability of a 6-game series is 31.25%. But what about a series that goes the distance, to a full seven games? Surprisingly, the chances of this happening are also exactly 31.25%.
This peculiar equality arises from the symmetry of our 50/50 assumption. For the Celtics to win in seven games, they must either win two out of the first five and then win the final two, or win three out of the first five, lose game six, and then triumph in game seven. Each of these scenarios has half the probability of winning in six games, but since there are two paths to victory, they sum to the same 31.25% chance.
Putting it all together, we find that there is a staggering 62.5% chance of the NBA Finals lasting six or seven games. The expected value, calculated by summing the products of each scenario’s probability and the corresponding number of games, is 5.81, which rounds to 6.
This fascinating result holds true even when we adjust the win probability for the favored team. Whether the Celtics or Mavericks are given a 55%, 60%, or 65% chance of winning each game, the expected length of the series still rounds to six games. Only when the probability becomes extremely lopsided, exceeding 66%, does the expected value drop to five games — and it’s exceedingly rare for such an unbalanced match-up to make it all the way to the NBA Finals.
As you can see from the chart below, the finals series from 2000 to 2023 align reasonably well with this back-of-the-envelope estimation, just with a few more 5-game and 6-game series at the expense of 7-game thrillers. But taking into account that the teams are rarely evenly matched, home court advantage is small but real, and there’s always a bit of noise in data, reality seems to match nicely.
So as you watch the Celtics and Mavericks, appreciate the spectacle not just for the basketball, but for the underlying mathematics that make each game a part of a larger, fascinating pattern.
No matter how the series unfolds, the math tells us that we’re likely in for a six-game spectacle. The NBA Finals have a way of conforming to the captivating logic of probability, reminding us that in the unpredictable world of sports, there’s a certain comfort to be found in the reliable embrace of mathematics.
In a world that often feels chaotic and unpredictable, it’s remarkable to discover that even something as exciting and emotionally charged as the NBA Finals can be understood, at least in part, through the lens of probability. This realization not only deepens our appreciation for the game but also invites us to look for the hidden patterns and underlying structures that shape our lives. By recognizing the role of probability in sports, we can begin to see its influence in other areas — from the stock market to the weather, from the spread of diseases to the outcomes of elections.
In doing so, we gain a newfound respect for the power of mathematics to help us make sense of the world, and a renewed sense of wonder at the complex interplay of chance and choice that defines our existence.
The author, Michael Bagalman, has been interested in sports statistics since the late 1980s. He clearly remembers writing to the Boston Celtics to ask for an unpaid internship for the summer of 1990 to let him analyze data for them. They said, “No.” Eventually in the late 2000s he finally got to work for an NBA team when he analyzed season ticket data for the Indiana Pacers to help with marketing efforts.
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