Watching the final moments of a close basketball game can be quite a thrill. When one team leads the other by only a point or two, one well timed shot at the buzzer can turn a bittersweet loss into a triumphant victory. Such was the case, for example, in this 2010 game between the Cleveland Cavaliers and the Utah Jazz.
A three point shot by Sundiata Gaines turned a two-point loss for the Jazz into a one-point win. No doubt that's a tough defeat for Cavs fans and players alike, but in such a situation, there's really nothing the defense could've done to change the outcome.
Or is there? What if, instead of letting Gaines take the shot, the defense had fouled him? Could that have increased the Cavs' likelihood of maintaining their lead? If Gaines had been fouled he would've been given three free throws, but would've had to make all three in order to win. Making three shots certainly sounds harder than making one shot, even if a shot from the line is easier to make than a three-pointer. Though ethically murky, is fouling a sound strategy mathematically?
To answer this question, the defense needs three pieces of information. First, regarding the player making the shot, they need to know (1) his free throw percentage, and (2) his three-point percentage. Without these numbers, they can't possibly determine whether it's better to foul or not. For Gaines, his career free throw percentage is 56.2%, and his career three-point percentage is 30.1%.
Already this is enough to determine the probability of a defensive loss if they decide not to foul: 30.1%, since they lose precisely when the three point shot is good. But if the defense does foul, they can lose in one of two ways: if Gaines makes all three foul shots, or if he makes two out of three and the defense loses in overtime. Therefore, the defense also needs an estimate of (3) the probability that they will lose in overtime, which we'll call P(D loses in OT).
Using Gaines' free throw percentage, and assuming the shots are independent, the first outcome (making three shots) occurs with probability .5623, or around 17.8%. The second outcome is a little trickier, because there are three ways Gaines can make two out of three foul shots (make/make/miss, make/miss/make, miss/make/make). Since the probability of making a free throw is 56.2%, the probability of missing one is 43.8%, and so the sum of the probabilities of these three cases equals
3(.562)2(.438) ≈ 41.5%.
This is only part of the story, since we still need to factor in the probability of an overtime defensive loss. The total probability in this case is 41.5% x P(D loses in OT) (more on why we multiply below). Therefore, if the defense fouls, they will lose with probability 17.8% + 41.5% x P(D loses in OT).
Before we drown in calculations, let's step back and ask what we've done. We have two values for the probability that the defense will lose: one if they foul, and one if they don't. The defense should do whatever minimizes their probability of a loss. This means they should foul only if the probability of a loss decreases by fouling. To phrase it mathematically, they should foul only if
17.8% + 41.5% x P(D loses in OT) < 30.1%.
If we isolate P(D loses in OT), we find that it must be less than around 29.6%, which doesn't seem very reasonable - after all, if the defense was really that much better than the offense, it's unlikely the game would be so close at the end! It looks like the Cavs did the right thing by not fouling.
Now let's ask a more general question: is it ever smart for an NBA team to foul at the buzzer when they are up by two and the opposing team goes for three? Suppose the shooting player's three point percentage is q and his free throw percentage is p. As we saw in the previous example, the defense should foul when
P(D loses given D fouls) < P(D loses given D doesn't foul).
P(D loses given D doesn't foul) = q, since the defense loses in this case precisely when the shooting player sinks the three-pointer. If the defense does foul, we have
P(D loses given D fouls) = P(no OT and D loses) + P(OT and D loses).
The first probability on the right hand side is p3 (we've replaced 56.2% in the argument above by a general p). For the second term, by definition of conditional probability we have
P(OT and D loses) = P(OT) x P(D loses in OT)
= 3p2(1-p) x P(D loses in OT).
P(OT) = 3p2(1-p) by the argument in the specific case, again replacing 56.2% by p.
Pulling all this together, our original inequality governing when fouling makes sense takes the form
p3 + 3p2(1-p) x P(D loses in OT) < q.
It seems reasonable to assume the teams are fairly evenly matched (otherwise it's unlikely the game would be so close at the end), so let's consider three cases: P(D loses in OT) = 40%, 50%, or 60%. By specifying this probability, we now have an inequality involving only two variables: p and q.
Now let's return from the heady land of mathematics to the concrete world of the NBA. We've already seen that Gaines should not have been fouled in this situation. Can we find players who it would make sense to foul?
To explore this question, consider player data from the 2011-2012 NBA season. In the scatter plot below, each point represents a player. The x-coordinate gives his free throw percentage (FT%, or p), while the y-coordinate gives his three point percentage (3PT%, or q). Because this season was shorter than is typical, for some players there isn't much data. Blue data points correspond to players whose numbers are less statistically significant (they attempted fewer than 20 free throws or three pointers), while red data points correspond to players whose numbers are more significant (they attempted at least 20 of each type of shot).
Now let's plot the graphs of p3 + 3p2(1-p) x P(D loses in OT) for different values of P(D loses in OT). We have already seen that it makes sense to foul a player if his three point percentage is larger than p3 + 3p2(1-p) x P(D loses in OT) - graphically, this means a player's data point should lie above the graph of the curve. Here's the same plot with the graphs superimposed, corresponding to P(D loses in OT) = 60%, 50%, 40%, and the extreme case 0%.
What can we learn from this picture? First, even with a 50/50 chance of winning in overtime, it doesn't look like it ever makes sense for the defense to foul. As P(D loses in OT) decreases, so too does the corresponding curve, but fouling never becomes a particularly attractive option. For instance, when P(D loses in OT) = 40%, there's only one player whose corresponding red dot lies above the orange curve. That player is Chandler Parsons of Houston (FT% = 55.13%, 3PT% = 33.71%), and even for him, fouling just barely makes sense. By fouling him, the defense's probability of losing goes down from 33.71% to 33.11%.
Also, even if the defense is guaranteed an overtime win (i.e. P(D loses in OT) = 0%), the majority of red dots in the scatter plot lie below the dashed curve. No matter how good the defense, it just doesn't make sense to foul most NBA players in this situation.
In conclusion, fouling rarely makes sense, though as with so many things in life, one should never say never. Can you think of a type of player who it might make sense to foul? And what if the defense had been up by one instead of two? Three instead of two?
Teachers: interested in discussing this topic with your students? Then check out the "Three Shots" lesson for an investigation of these ideas in a slightly different context!