Should You Foul When You’re Up By Three?

To foul or not to foul -- that is the question. There are few debates that rouse greater in hoops circles than whether to foul in the closing seconds when up by three points. At first blush, fouling would appear to be the better bet. After all, rather than equalizing things with just a single shot, a team has to make the first free throw AND get a hold of the rebound off a deliberate miss AND make a shot (although, as we shall see, there are other ways this could play out). Intuitively, it’d seem that as a coach you’d prefer the course that makes the other team string a slew of low likelihood plays together.

Aside from worries about pulling it off without fouling during a shot, there’s also the fact that fouling does open up the possibility of losing in regulation, albeit a minuscule one. But for a risk-averse profession like NBA head coaching, any increased probability of blowing a lead can be enough to make them forswear an unconventional stratagem. So who’s right? Is this a case of new-school sabr-guys taking on the entrenched “wisdom” of the old school? Or is it more complicated than that? (Warning: What follows is a math-intensive exploration of the various probabilities. If you don’t feel like getting your nerd on, skip below for the conclusion).