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The launch of the recent nbatop.com Research Program has brought us a flurry of very interesting ideas for NBA analytical articles, many of which we hope will eventually find their way to publication on the site.

"3-D": Late-Game Defensive Strategy with a 3-Point Lead

NBA Statistical Analyst Adrian Lawhorn

by Adrian Lawhorn

We've all been there. Our team has a 3-point lead, and the clock is winding down. The opposing team has the ball, and it looks like we're one play away from victory. What should we do defensively to secure the win?

The purpose of this research article is to answer precisely that question. First, let's take a look at the most definable situation: when the game clock is down to a few seconds (final possession), and the team with the ball needs a '3' to tie the game and send it into overtime.

The league average for 3-point FG percentage (last full season) is 35.5%; however, late-game shots are known to have a higher degree of difficulty than the typical 3-point attempt. To do this analysis accurately, we need to find out what the 3-point FG percentage is in these exact situations...the following table contains the answer, using data from the past two seasons:

Time Left


3-Pt FGM

3-Pt FGA

FG Pct

<11 Seconds 3 Points 41 205 20.0%

Therefore, with 10 seconds or less left in the game, and a 3-point deficit for the offensive team, the league-wide 3-point shooting percentage is 20.0%. This should be a large enough dataset to give us an accurate read on the situation...so, if our team chooses to defend the 3-point line in this scenario, these are the odds that we will be forced to play five more minutes of basketball, rather than walk off the court with a 'W'!

Now, what if our team chooses to foul? In order for this to play out in a favorable way for the trailing team, several things need to happen:

- The fouled player must make the first free throw.
- The fouled player must then miss the second attempt on purpose.
- The trailing team must then get an offensive rebound off the FT miss.
- The trailing team must then score off the rebound.

The big question is, "what is the probability of this happening?". Looking at data from the last couple years, we find that there were 32 occurrences of this scenario. Of those, the trailing team was successful 3 times...a percentage of 9.4%. Now, this is not very conclusive because the sample size we have is very small. If we had a much larger data sample, would that percentage go up, or would it go down?

So...are we stuck? Not at all! Fortunately, this is something we can calculate without too much trouble. Using Probability and Statistics principals, we can multiply the individual percentages of all the events above to determine what the percent chance is of these events happening in this order. And the numbers are...

- The fouled player must make the first free throw: 75.6%
- The fouled player must then miss the second attempt on purpose: 100.0%
- The trailing team must then get an offensive rebound off the FT miss: 13.9%
- The trailing team must then score off the rebound: (see below)

These stats are arrived at by using league averages. Although it is not unheard of that a player will actually make a FT when trying to miss, we'll assume here that a player can execute an intentional miss each time they try. Also, the percentage for scoring off an offensive rebound can be broken down two ways: put-backs (50.4%), or the ball is kicked back out by the rebounder to a teammate (42.2%). So, now we'll do the math...

(.756) * (1.00) * (.139) * (.504) = .053

(.756) * (1.00) * (.139) * (.422) = .044

There you have it...if our team fouls in this situation as the clock is running out, there is a 5.3% chance that the opposing team will force overtime with a put-back off the offensive glass, or only a 4.4% chance that the opposing team will force overtime when the rebounder passes the ball to a teammate. Now, let's summarize the findings so we can draw some conclusions:

Time Left

'O' Deficit

'D' Strategy


Overtime Pct

<11 Seconds 3 Points Foul Score on put-back 5.3%
<11 Seconds 3 Points Foul Score on kick-out 4.4%
<11 Seconds 3 Points No Foul Defend the 3-pt line 20.0%

The basic conclusion we come to is this:
With time running out (final possession), and a 3-point lead, the defensive team is roughly 4 times more likely to "blow the lead" and have to play overtime if they choose to play defense, rather than foul in the waning seconds of the game.

This is a satisfying conclusion, but what about some of the other possibilities? A few words about that...

I don't think anyone would suggest that intentionally fouling a 3-point shooter in the "one possession" situation is a good idea, but let's see how that holds up statistically. If the player fouled in this case is an "average" NBA free throw shooter (75.6%), we can calculate the odds that he will make all three free throws and send the game into overtime:

(.756) * (.756) * (.756)  = .432

Not that this is a surprise, but we would not recommend that the defensive team foul a 3-point shooter on the last possession, since the odds of him hitting the free throws and forcing overtime are relatively high (43.2%)!

Now, a look at what might happen when there is enough time left on the game clock for multiple possessions...

Because there are so many permutations of events that can occur when each team has the ball two or more times, it would be outside the scope of this article to do a study of every possibility. However, we can make some general statements about this case. When there is enough time left, some teams will go for the quick '2', then foul the opponent right away after the inbounds pass. Others may attempt a '3', and hope for the tying shot to go in, or at least try to get an offensive board off the miss, and then try again! So what can we say about this? The best thing we can do is look at our original data sample, and see what we've got...

As mentioned earlier, we found 32 cases where our base scenario was met (and the defensive team fouled) in the last two NBA seasons. When we include all outcomes (one possession, as well as multiple possession sequences), we find that the team with a 3-point deficit was able to tie the game 7 times...a percentage of 21.9%. As we've discussed, this is a small (inconclusive) sample size, but at least we can get a little information from this. Based on the data we have, we cannot make a blanket statement as to the best strategy for the multiple-possession scenario...the odds of success when fouling vs. not fouling are in the same range, according to our limited information.

Lastly, a note about turnovers. Yes, turnovers can occur in these situations. The reason we haven't really covered that here is two-fold: turnovers are not common in this scenario, particularly when you throw out things like bad pass turnovers that are "desperation" passes, many times from the backcourt, as the clock is running out. Also, to do this correctly, we'd need further information about how and why these turnovers took place - which would require some in-depth game charting information that we currently don't have available.

So, whether you are a fan of the game, or you own, manage, or coach an NBA club, keep this in mind: the next time your team finds itself up 3 points, and you need to make a final play on defense to win the game - think "3-D" strategy!

Adrian Lawhorn has a B.S. in Mathematics from Rose-Hulman Institute of Technology, one of the top-rated undergraduate science and engineering school in the country, and is the Founder/Owner of Wizard Sports. He has been working as a professional sports statistician since 1996, and covering the NBA since 1997. You can visit his website at: http://www.adrianlawhorn.com/

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