Shooting yourself out of the game? The 3 pointer and the Binomial Distribution.

The evening after Thanksgiving saw the Houston Rockets up to their normal tricks. Specifically, shooting 3 pointers. Lots of them. 37 it total. Nearly half (45% to be exact) of every shot they took that night versus the visiting Clippers were from behind the 3 point arch. Unfortunately, on this night they weren’t falling for the team that leads the league in 3 point attemps by a wide margin.

Houston is currently in the vanguard of the 3 point revolution in the NBA. Since the 3 point line was introduced in 1980, the number of 3s per game has been steadily rising, from a scant 2.8 attempts per team per game to over 21 per game last year. This flies in the face of the conventional wisdom that 3s are a fickle friend at best. “Win by the 3, die by the 3.” 3 point shooters run hot and cold, the traditionalists would say, and 3 pointers have been considered hard to rebound.

But stat heads like Houston’s GM Darryl Morey might rightly point out that a 3 pointer is worth more than a 2 if made. Last year, the league average makes for 2 point attempts is ~48% while the league average for 3 pointers ~36%. There is profit in those margins!

So is there anything to the conventional wisdom? Well there is something. Consider flipping a coin 10 times. On average you’d get heads 5 times, but that’s just the average. Sometimes you’d get 3, sometimes you’d get 6. Exactly how many times is described by something called the binomial distribution, which is somewhat related to the Bell Curve most of us are familiar with from being graded most of our youths.

Well 2 and 3 pointers also have distribution, in fact they can be approximated by the binomial distribution. Below is the binomial distribution for a 100 coin tosses, or makes out of a 100 2 point attempts if you shoot 50%.

3 poiters have a similar, but not identical binomial distribution. The 3 point distribution is a little asymetrical and pushed off to side. Is this important? Well yes. Let’s run a quick experiment.

Assume that at the start of the 4th quarter, two teams, let us say Houston and the Sacramento Kings, are tied. For the duration of the 4th quarter, Kings only take 2 pointers, 30 in total, and make 50%. On average, assuming no free throws for simplicity’s sake, the King’s would hit 15 shots and score 30 points.

Meanwhile, let us assume that the Rockets only take and score off of 3 pointers. They make 40%, but only attempt 25, presumably because taking 3 pointers puts the Rockets at a rebounding disadvantage. (The Rockets are currently 26th out of 30 in rebounding.) On average, the Rockets make 10 3s, also for 30 points. Even, right? Well no!

You still have to take the shots and flip the coins and see where everthing lands. Let’s assume the Kings and Rockets play out this scenario a million time. Who comes out on top more often, is it a 50/50 proposition, as one might intuitively think? Thanks to statistics libraries like R, this is be done with next to no pain.

x<-rbinom(1000000,25,0.4)*3 #repeated a million times, 25 three points attempted, 40% 3p, each make is worth 3 points

> y<-rbinom(1000000,30,0.5)*2 #repeated a million times, 30 two points attempted, 50% fgp, each make is worth 2 points

> sum(x>y)/1000000

= 0.474648

Wow! The Rockets only win in this scenario about 47.5% of the time. That’s worth a couple game over an 82 game season. Even though on average over those game, the teams would score the same amount in the 4th quarter. In an NBA were team execs are crunching the numbers trying to maximize the outcome of every possession, this is a huge result. How is it even possible? Well in plain terms, yes, 3 pointers run hot and cold. More so than 2 pointers at least.

Now before the statistically inclined start berating me for being simplistic, let me say, this is a first order demonstration… at best. It is simplistic. For one, we can’t just consider the points distribution of the fourth quarter. If we are going to play the distribution game, we should apply distributions to the first 3 quarters (i.e., say that on average after 3 quarters the Kings and Rockets are tied, and the actual score for a given trial is determined by a given distribution….).

For another, why presume that the Rockets get 5 less rebounds? Yes, this year they are 26th in rebounding, but that is at least in part due to the fact that Dwight Howard has been injured. Last year they were 4th in the league in rebounds.

Further, is a distribution for a 2 or 3 point shots made really described by a binomial distribution? Well no. The actual distribution is certainly more complex, but it isn’t unreasonable to assume that a binomial distribution is a good approximation.

And there are more caveats and objections, to be sure. But, the underlying point is sound. There is, as the old guard who scoff at the Moneyball nerds contend, a real chance that the inherent streakiness of 3 pointers could shoot a team out a game.

So is the Moneyball crowd wrong then? Did they get suckered by the allure of easy pointer from beyond the arch? No! The effect of the distribution is real, but it can be more than offset simply by rebounding threes effectively, and Darryl Morey and the Houston staff knows this I’m sure.

The most interesting thing about that Rockets/Clippers game was a comment by Clippers announcers and former NBA player Mike Smith. He said that before the game he had a chat with Rockets coach, Hall of Famer Kevin McHale. According to Smith, McHale told him that during the off season the Rockets reviewed game tap and identified 5 areas on defense where they could push the envelope of the rules more. Understandably, Smith didn’t say on air what those five area of not-quite-rule-breaking were, but it was a fascinating insight into an organization which approaches basketball as a problem to be solved.

The Rockets players may have been overladen with Turkey in their loss to the Clippers the day after Thanksgiving. They only made 10 of 37 3 point attemps (27%). But so far this season they are doing something right. In spite of Dwight Howard’s being injured and losing out on several free agents in the off season, the Rockets are 14–4 to start the season.