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*?

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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 .562^{3}, 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.

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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 *p*^{3} (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)

= 3*p*^{2}(1-*p*) x P(D loses in OT).

P(OT) = 3*p*^{2}(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

*p*^{3} + 3*p*^{2}(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 *p*^{3} + 3*p*^{2}(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 *p*^{3} + 3*p*^{2}(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!

As a former basketball coach (and current math teacher), I never found fouling at the end of the game ethically wrong but rather a strategic move for some players. I love the above analysis and math, but have a couple of points that could also be considered. At the end of a game with the outcome on the line, the pressure is significantly more than at other times in the game, so this might have an impact on a player's free throw percentage. Don't know if that can accurately be measured, but it's something to consider. Second, at least in high school and college, teams are usually in the bonus, so if you foul before they start the act of shooting a three point field goal, they will only get two shots. So that is something else that a coach must consider. Great post, as I don't think people always realize how sports, strategy, and math cab be in such a relationship.

Great point about nerves. Wouldn't the three-point percentage also be lower for the same reason?

Probably, but there is just something about that free throw line at the end of the game. Maybe it's because the flow of the game has stopped, at the line by yourself, etc. BTW, you guys are killing it. Keep it up.

Flow means a lot. There are guys who can stand the pressure for one shot (the 3) but the pressure mounts in the sequence of 3 foul shots. This is great fun. Thanks for this.

Money Ball. You can make a career out of this.

Good article but you did leave out a possibility. You are assuming he is getting three shots or in the act of shooting. There is also the possibility he is fouled before he can get the shot off and there is also the possibility that even though he is fouled he still makes the shot.

I think you could take nerves into account if you used more situational shooting statistics. What is a player's FT% with the game on the line or in the last 30 seconds of a game? Same can be done for the 3pt shot. And then the scatter plot and graphs could also be changed to reflect this data league-wide so it could be seen if more players fall above the curve. It might be possible that a seam of 'chokers' could make fouling on a game winning 3 a rarely used but statistically sound strategy.