The REAL Trouble With Foul Trouble


Using play by play data from the 2020 NBA season there are identifiable instances of coaches overreacting to foul trouble. Only a small percentage of players foul out of games because they are intentionally prevented from being put in the position to foul out even though there is no penalty for fouling out besides being disqualified from the game. Coaches do this so they can have their best players late in the game when the outcome is most clear, but risk taking out a player out for too much time putting their team at a big disadvantage.

  • Big Men
  • Role Players


There is an unwritten rule in the NBA that if a player gets into foul trouble you must take him out immediately without a second thought. Right there on the spot. It doesn’t matter if it’s your best player or the last player off the bench.
  • There are times where a player was not playing in a game, a quarter ended, and then they had an event in the game (scored points, made an assist, got a rebound, etc). We are going to assume they were substituted in during the break between quarters

Establishing Some Ground Rules

What is foul trouble?

  • Has at least three fouls in the first half of a game
  • Has at least four fouls in the first three quarters of a game
  • Has five fouls in the fourth quarter
This bar chart shows the percentage of time a player gets immediately taken out after a foul. Blue represents players who either became or were in foul trouble on the foul, red represents players who fouled but did not become in foul trouble

How often does a player foul out?

There might be high urgency to prevent players from fouling out, but is the paranoia legitimate? There would be cause for concern if six fouls in one game were too few for players to be comfortable. If that was the case, we would see a high frequency of players foul out over the course of the season. The figure below shows that is not the case.

The image above shows the percentage of players who ended up fouling out of the game based on the quarter of the game as well as the number of fouls they had. The percentages are all low, showing a player fouling out is not a common event.

Assignment of Players

All NBA players are not created equal. There are five positions on the court, but over the years the NBA has seen a transformation of five unique type of players mold into players who can fit any position. We’re now seeing big men, who used to just sit under the basket, pull up and shoot from 25 feet away from the hoop. A game where the fundamental idea used to be working the ball inside has shifted to see how often you can launch the ball from the three point line. Point guards now play isolation basketball with the intention of getting fouled to shoot free throws. It’s a whole new game.

This image shows the breakdown of the median percentile for each cluster for different categories
  • Cluster 2 — Big Men. These players have high field goal percentage and high two point percentages to go along with their high rebounds per minute. There are 71 players in this grouping.
  • Cluster 3 — Role Players. These players have percentiles that are mostly in the 30% and never go higher than 60%. They come off the bench and fill the gaps when necessary. There are 178 players in this grouping.

Fouls Per Minute Meets the Poisson Distribution

Now that players have been assigned to different clusters, what can that tell us about how coaches should determine whether to keep or leave players in a game based on the number of fouls they have? The main idea is to determine the expected number of minutes a player would have played on any given night and the number of fouls per minute for that player to predict the probability they foul out of the game. If the probability of them fouling out of the game is above a threshold, they should be allowed to stay in the game. If it is below the threshold, they should be kept out of the game until the probability becomes lower than the threshold.
The probability density function of a Poisson distribution with lambda equalling 2.4 and X equalling four
The probability of fouling out of a game when a player has 2 fouls in a game and expects to play another 24 minutes with a foul rate of one foul every ten minutes
Distribution of fouls per minute of players in each cluster
  • Cluster 2: .11 fouls per minute
  • Cluster 3: .08 fouls per minute

When A Player Should Actually Be Pulled

We’ve talked a lot about why coaches are too reactive to taking out their players to prevent fouling out, but we haven’t given the answer of when it is wise to take players out of the game. Using foul data from the 2020 season and assigning players to their cluster, the visualization below walks through the likelihood of a player fouling out of a game in different scenarios.

This line graph shows the likelihood of a player fouling out if they were to play X more minutes depending on the number of fouls they have in the game as well as the cluster they belong to. Cluster 1 is a solid line, Cluster 2 is a dashed line, and Cluster 3 is the dotted line.


There should be no reason why there are so few games where a player fouls out. Coaches seem to be in constant fear of an outcome that has no real punishment. The idea that time at the end of the game is more valuable than at any other point in the game is an illusion. The reason why coaches want their best players on the court at the end of the game is because the outcome is easier to see than at any other point, but sacrificing playing time of those players early in the game puts teams in worse position at the end of the game. Understanding foul trouble management gives coaches the ability to prevent hitting the panic button too early and putting their team in the best position to win the game.



The Elbow method to determine the number of clusters to pick for the K means clustering algorithm. This method is common in choosing the right number of clusters by determining the total within sum of squares (TWSS) between points and their centroid. Once K begins to increase, the TWSS slowly becomes smaller and smaller. The idea is to choose the smallest value of K before the change in TWSS doesn’t give more information about the centroid’s location. In this case, 3 was the number chosen.

The elbow method that was used to choose the number of clusters for the k-means clustering algorithm.

Poisson Distribution

The Poisson distribution has some properties about it making it a great choice for calculating probabilities with foul trouble.

  1. It has a memoryless property. This means it doesn’t matter how long its been since the last event, it is only calculating based off the current time. This is great for this because the 2nd foul for a player shouldn’t have an impact on their 3rd foul (excluding intentional fouls at the end of the game)
  2. The Poisson distribution works with discrete events. You can’t have 3.4 fouls in a game, you can only have discrete values.

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