On March 3, 2014, Kyle Korver took five three-points shots against the Portland Trailblazers and missed all of them. This marked the first time in two years that Korver had played in a regular season game without making at least one three-point shot. Korver has played in over 790 regular season games in the NBA, but his previous streak for games with at least one three-point shot made was only 28. Clearly, Korver’s 127 games streak is remarkable, but how likely was it? One method for understanding the likelihood of the streak is to simulate the chances of Korver making a three-point shot in each of the 127 games. Given Korver’s long history in the NBA, we can use his career statistics to inform our simulation of the streak. What follows is the setup and results from our simulation; we also offer potential extensions to this analysis, whose completion is truly a function of the level of data available and the level of effort a researcher wishes to expend.
While we know that Korver hit at least one three-pointer per game during the streak, we’d like to know what was the probability of him hitting at least one three-point shot in each game. In order to do that, we first need to model the number of three point shots attempted in each game. We assume that the number of three point shots taken in a game can be modeled using a Poisson regression. This type of regression is common with count (non-negative integer) data. We model the number of three point shots attempted in each game as a function of factors such as whether the game is at home or away, minutes played by Korver, the record of the opponent, whether the Hawks are in playoff contention, etc.
Once we have estimated the number of three pointers attempted in a game, we simulate the probability of making at least one three pointer using the binomial distribution. The binomial distribution provides the probability of k successes over n trials, where the probability of success in each trial is p. In this context, k is a made three-pointer, n is the number of three point attempts (estimated from the Poisson regression), and p is sampled from a normal distribution based on Korver’s historical three-point percentage mean and standard deviation. The binomial distribution assumes that the probability of hitting a three point attempt in a game is not connected to if the shooter hit or missed his last three-pointer (there is independence across trials). We can express the probability of making at least one three-pointer as:
We then multiply these 127 game probabilities together to compute the overall probability of Korver’s streak. By taking the product of the individual game probabilities, we are assuming that they are independent. There have been several arguments for why there is no “hot-hand” while shooting within a given game, thus we don’t feel it is unreasonable to assume that there is no “hot-hand” across games (although the “hot-hand” can be easily incorporated into our model).
Now, we are ready to run our simulation. For all 127 games of the streak, we simulate the probability of making at least one three-pointer. We then multiply these simulated probabilities together to obtain the overall probability of the streak. We perform this exercise 500,000 times to get a better understanding of the simulated overall probability of observing Korver’s streak. Figure 1 is a plot of these 500,000 simulations of the overall streak. The average of these simulated probabilities is just 0.0000000003843 (where 1 = 100%)!
If you are anything like us, your head is already full of criticisms of our approach. Let us address these criticisms through potential extensions of our simulation. First, it is possible to relax the independence assumptions (both within and across games) if you believe that there is a “hot-hand” in basketball. Second, of course, the ideal Poisson model of the number of three-point shots attempted would also employ play by play in-game data, where we would observe the score differential, the time remaining, the defense being played, etc. These in-game situational factors would help determine if Korver launched a three-pointer. Such an analysis would require access to in-game data and a great deal of time and resources.
The longest current streak for at least one three pointer made in a game is 51 by Stephen Curry of the Golden State Warriors. In order to tie Korver’s streak, Curry would have to make at least one three pointer in each of his next 76 games. We decided to simulate the probability of Curry tying Korver’s streak. Once again, we estimated the number of three-point attempts for each game using a Poisson regression. We had to limit the covariates in the regression, since we are projecting into the future. We also truncated the regression to guarantee that Curry attempted at least one three-pointer per game. We then used the binomial distribution to simulate the probability of hitting at least one three pointer given the estimated number of three-point attempts. We took the product of these 76 games to determine the overall probability of Curry tying Korver’s streak.
Figure 2 is a plot of 500,000 simulations of Curry tying the overall streak. The average of these simulated probabilities (0.000006281) is more than 15,000 times that of the probability of observing Korver’s streak! This reflects not only Curry’s prowess as a three point shooter, but it also shows the true exceptionality of Korver’s accomplishment (please take note, Mr. Rovell).
Mike Lewis & Manish Tripathi, Emory University, 2014.