When to Foul the Shooter

The conventional basketball wisdom is to foul the shooter rather than give up an easy lay-up. If a shooter has an easy shot, say a shot he makes 95% of the time, it is better to foul him and let him attempt two free throws rather than take the easy shot. As long as he does not make over 95% of his free throws (which almost nobody does), the defensive team will allow less expected points. If the offensive player only makes 50% of his free throws, fouling him saves .9 expected points. 2*.95 - (1*.5+1*.5) = .9 . Fouling the league average shooter, who makes 75.2% of his free throws saves about .4 points. Fouling has another detriment that is not accounted for in that math. Once a team has commited five fouls, the other team goes to the free thow line for every subsequent defensive or loose ball foul, regardless of whether the player was in the act of shooting.

During the 2006-2007 NBA season, teams scored about 1.1 points per possession and made 75.5% of free throws. If the offensive team is in the bonus, a non-shooting foul results in the defense allowing .43 more expected points than they do on an average possession. 2*.752 - 1.1 = .43. It is better to let the possession elapse without commiting a non-shooting foul. The defensive team would rather not be in the bonus. Should you still foul the shooter on an easy shot?

Before estimating the additional penalty that committing a foul detracts, here are some statistics:

Free throw % and the other stats needed to calculate Points Per Possession and Mean Fouls were found here. Points Per Possession was found by taking the league average Points Per Game, and dividing by league average 'Pace' statistic. PPG/Pace = PPP. I calculated Mean Fouls by taking the league average minutes per season, divided by 5 to make the stat minutes per team, and divided it by the league average fouls per team to get team fouls per minute. I multiplied it by 12 (minutes in a quarter) to get fouls per quarter. (Minutes/5)/(Fouls per Team) * 12 = FPQ.

I will take an extreme case to see if there is an instance where it would be better to let the offensive player have any easy shot, rather than foul. Suppose that the offensive player has an easy shot at the start of the quarter. Should he be fouled? The Poisson Distrubution, can be used to show how often a team reaches a certain number of fouls.

In this case λ = 5.51 (average fouls per quarter) and k = fouls in a quarter. On the first line of the table below is the random variable k, ranging from 0 to 14. The percentage below it the Poisson probability of that exact amount of fouls happening in a quarter. For example, the most likely outcome is a team commiting 5 fouls in a quarter, which happens 17.1% of the time. Below that number is 'Points Lost.' Points Lost is the amount of points the defense loses by fouling a player and letting him shoot free throws. As shown above, by fouling the defense allows .43 more points than they would if it did not foul. Multiplying it by the Poisson probability gives the points lost. For example, by fouling 5 times in a quarter, on the 5th foul the team goes to the free throw line and gets .43 more points. This happens in 17.1% of quarters. The penalty increases as the team fouls more often. Fouling 6 times a quarter send the opposing team to the free throw line on two occassions, allowing the offensive team to get .86 more points per quarter. The formula is .43*Poisson%*(k-4) = Points Lost. 4 being used because every time a team commits k > 4 fouls the opposing team goes to the free throw line k-4 times. Total Points Lost is the sum of expected Points Lost for each value of k.

To return to the extreme case, fouling at the beginning of the quarter has the effect of the offensive team needing to draw only 4 more fouls (rather than 5) in order to shoot free throws. Compared to the previous example, this increases the Total Points Lost by the defense. Committing 4 fouls results in Points Lost and the penalty for committing more fouls increases. For this case, Points Lost = .43*Poisson%*(k-3).

If the difference between Total Points Lost (Early Foul) and Total Points Lost (normal case)increases by more than .4 points, the amount of points the defense saves by fouling the league average player on an easy shot, then the defense would be better off not fouling. The table below shows the comparison:



The difference in Total Points Lost is .343 which is < .4. Although the defense's Total Points Lost increases with the early foul it does not increase enough to justify letting a player have an easy shot. Thus the conventional wisdom is reaffirmed for the league average player; foul him rather than let him have an easy shot. If Total Points Lost had been more than .4, the next step would have been to calculate a more accurate Total Points Lost by accounting for offensive fouls (which do not result in free throws) and shooting fouls (which always result in free throws). However that is not needed and the reader will never learn that 9.8% of all fouls committed during the 2006-2007 NBA season were offensive fouls.

While all I did was reaffirm the conventional wisdom, I believe the data is suggests that there may be special cases when the defense should not commit the early foul. If the offensive player makes easy shots somewhat than 95% of the time and makes free throws somewhat more than 75% of the time, it may be worth not commiting an early foul on him. NBA teams with access to more exact stats would be advised to, especially during 7 game playoff series, calculate some exact figures for specific players to see who and when not to foul.