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.

Doc Rivers is one of the evangelists of late-game fouling in this situation, as this New York Times piece chronicles, although the verdict is hardly unanimous amongst his colleagues on the sidelines. Because as simple as "foul the guy before he can shoot it" sounds, in practice it can be much, more difficult. Just ask Antoine Wright.

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).

Let's try attaching some probabilities to these scenarios. For the sake of argument, let's say there are ten seconds left in a game, Team A is up by a trey and over the foul limit, Team B has the ball, and both squads have a single timeout left. What should Team A do? Let's run through all of the conceivable possibilities and calculate the chance that Team B wins, given by W (and remember, overtime is usually a coin flip -- not literally like in the NFL, but figuratively speaking -- so in any of the the below scenarios where the game would go to an extra period, the chance of winning is the chance of forcing overtime multiplied by half).

• Case 1: Team A eschews the foul and Team B attempts a three-pointer. So, W=3PT%*0.5
• Case 2: Team A fouls in a non-shooting situation, Team B makes the first free throw, misses the second, rebounds the ball, and makes a two-pointer to force overtime. So, W=FT%*OReb%*2PT%*0.5
• Case 3: Same as above, except Team B knocks down a three-pointer instead of a deuce. So, W=FT%*OReb%*3PT%
• Case 4: Team A fouls, then Team B unintentionally misses the first free throw, deliberately misses the second, grabs the offensive rebound and makes a shot from distance to tie it up. So, W=OReb%*3PT%*0.5
• Case 5: After Team A fouls, Team B makes both free throws and immediately fouls. We'll ignore the possibility of Team A making both shots, since that's merely iterative of our baseline scenario: i.e., we're back to Team B being down by three, except with less time on the clock and without a timeout now. Instead, let's look at what happens if Team A misses a shot from the charity stripe. Things get a little more complicated here because the order of when Team A misses dictates whether there's a rebound to haul in. But, we can rather safely assume that Team A will want to send everyone but the shooter back on defense, so we'll go ahead and just give the rebound to Team B, who promptly calls timeout. With the margin reduced to two points, Team A has no incentive to foul, so Team B . To recap: Team B sinks both foul shots, Team A misses one of theirs and Team B evens things up with a deuce at the other end, extending the game to overtime. So, W=FT%^2*2*FT%*(1-FT%)*2PT%*0.5
• Case 6: Same as above, except this time Team B drains a shot from three-point land to win in regulation. With a little tinkering, we get W=FT%^2*2*FT%*(1-FT%)*3PT%
• Case 7: Once again, Team A fouls, Team B makes both ensuing free throws and then fouls after the inbounds. But this time Team A misses both free throws. Team B rebounds the ball, calls time out, and then hits a shot to win. So W=FT%^2*(1-FT%)^2*FG%

This is getting pretty dizzying. But the general gist is: what's bigger, the first case, or all of the others combined? Fortunately, we can plug some numbers in relatively easily to get a decent grasp of the issue. It turns out that teams in this exact situation -- up by three with just a handful of ticks left -- allow only 20% three-point shooting. The reason this number is well below the season average of 35.5% three-point shooting is that defenders are keying in on shooters and giving up easy twos in this situation. So the trailing team will win about 10% of the time (again due to the fact that overtime is more or less an even shot) if the other team spurns the foul.

Using data and some educated guesswork, we can fill in the rest of our variables with decently accurate approximations. Turning to the incomparable 82games.com, we find out that the chance of getting an offensive rebound off a missed free throw is 13.9%, that the shooting percentage on a quick putback off an offensive board is 50.4% and that the average free throw shooting percentage in the NBA is 75.6%. This gets us most of the way there. The only holes we need to fill in are what the shooting percentages should be with very little time left off an inbounds situation (essentially, the percentages we should use for Cases 5-7). Well, the 20% three-point shooting for teams behind by three late in games we used above makes sense for Case 6, and using that as a baseline, it seems pretty reasonable to guess that teams will make 30% of their two-pointers in the same circumstances.

Voila, now we can crunch some numbers. And the results are somewhat surprising. Summing up the odds of Cases 2-7 (all of the ones that involve the winning team intentionally fouling to prevent a potential tying trey), we get a 16.1% chance Team B wins. In other words, the team that's ahead would lose 6% of the time more if they foul in this particular scenario. Now there are two massive caveats. The first is free throw shooting. If the team with the lead has the chance to foul a subpar shooter on the other team, and conversely has some guys who are automatic from the charity stripe themselves, then fouling is a much, much better strategy. Second -- and much more importantly -- is the timeout situation of the team that's behind. If they're out of timeouts, then trying to make both free throws is more or less eliminated as a plausible strategy; they'd have to go full court in presumably a few seconds to either tie or win. That more or less halves the odds of them pulling off the comeback, making the foul the better percentage play.

The bottom line is that context matters. A lot. There's no quick and dirty rule for when to foul and when not to, as the time remaining and timeout situation dramatically alter the odds. But this shouldn't be too surprising; this nugget from Rockets front-office guru Sam Hinkie should have given us a clue:

"It’s a complicated problem," said Sam Hinkie, Houston’s vice president for basketball operations. "Sometimes I’ve been convinced it was right to foul and disagreed, other times I’ve thought we shouldn’t foul and others disagreed."

If Darryl Morey's crew of stat heads can't come up with a definitive answer, what hope do mere mortals like the rest of us fare -- let alone NBA players?

So just remember, the next time a team faces this situation in the playoffs, they should definitely maybe consider fouling. It's complicated.