Stat Chat

Stat of the week: 1.047, the expected value of any Brewer field-goal attempt.

Each season since 2010-2011, three-point attempts per game have gone up in the NBA, reaching an all-time high this year at 33.7 (Basketball Reference, “NBA League Averages – Per Game,” 02.02.2020). Vassar’s women’s basketball team has mostly followed suit, increasing their threes attempted per game every season since 2014-2015, reaching a 10-year high this year at 20.5 attempts per game (Vassar College Athletics, “Women’s Basketball Cumulative Statistics,” 02.02.2020). Should the Brewers keep putting up treys like there’s no tomorrow?

Steph Curry is largely credited with jump-starting the three-point revolution in the NBA. He ranks third all-time in three-pointers made, and has played seven seasons fewer than the two players, Ray Allen and Reggie Miller, with more (Land of, “NBA All-Time 3-Pointers Made Leaders,” 02.02.2020). He made NBA executives question their long-held offensive beliefs—the conventional wisdom that said driving into the paint for an easy bucket was the best way to score.

Dani Douglas shoots from downtown. Courtesy of Vassar Athletics.

However, three-pointers are worth 50 percent more than a bucket in the paint. Using the simple idea of expected value, a team that shoots 30 percent from beyond the arc and 40 percent from the field otherwise would average .303 = 0.90 points for every three attempted and .402 = 0.80 points for every two-pointer attempted. Seemingly ignorant of this simple math, coaches and teams kept three-point attempts down since the three-point line was introduced in 1979, as low as 17 per game as recently as the 2006-2007 season.

Is Vassar keeping up with the revolution, or do the Brewers need to shoot more threes? This year, they’ve shot a 34.9 percent clip from beyond the arc and 48 percent from the field otherwise. That puts their expected value for threes at .349(3) = 1.047 and their expected value for two-pointers at .48(2) = .96. Indeed, it seems like they should continue to increase their three-point attempts. Yet, we have neglected to look at free throws in this experiment, and free throws are far more likely to occur on two-point possessions. The Brewers are making an excellent 74.4 percent of their free throw attempts this season, their best in five years. While their expected value for one free throw, worth one point, is .744(1) = .744, you get two attempts for being fouled while going for a two-pointer that you miss. The possible outcomes (with their corresponding probabilities) when the Brewers are given two free throw attempts are as follows: the probability that both are buckets is .744(.744) = .55, that one goes in is .744(.256) + .256(.744) = .19 + .19 = .38, and that neither shot goes in is .256(.256) = .07. Thus, the expected value of two free throws is .55(2) + .38(1) + .07(0) = 1.48. The Brewers have been fouled in about one-sixth of their two-point attempts—they have made 809 attempts without being fouled and have been fouled on around 127 shots, for a total of 936 attempts. 127 is about one-sixth of 936. Roughly, we can get a new expected value for two-point attempts by assigning probabilities for each possible outcome, free throws or not. We get probabilities of .167 and .833, respectively, since the probabilities must add to one and the probability of free throws is about one-sixth (.167). This gives us an expected value of .833(.96) + .167(1.48), which comes to approximately 1.047…the exact same as the expected value for threes!

It’s optimal for all of your shots to have the same expected value; this means you don’t need to take any more of either kind of shot at the expense of the other. This in mind, the Brewers should just keep doing what they’re doing. Shooting optimally and with a record of 13-6, the rest of the Liberty League should watch out for our Brewers!

[Correction: An earlier version of this article included calculations that omitted the necessary parentheses. The parentheses have been added.]

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