## Sunday, June 5, 2016

### The Role of Three Point Shots

The Role of Three Point Shots

Question:  Consider two basketball teams.   The first team takes 15 two-point shots and 5 three-point shots.  The second team takes 10 two-point shots and 10 three- point shots.  On average, both teams make 50% of their two-point shots and 35% of their two-point shots.   What is the expected number of points from the 20 shots for the two teams?   What is standard deviation of the number of points for the two teams?

Answer:    For both teams the points from two-point shots and three point shots are binomially distributed.

For player one denote Y as number of two point shots made and Z as the number of three point shots made where Y=B(n2,0.5) and Z=B(n3,0.35) where n2 is the number of two-point shots made and n3 is the number of three point shots made.

POINTS= 2 *Y+ 3* Z

E(POINTS) = 2 * E(Y)  + 3*E(Z)

Using the fact that Var(ax) is equal to a2 var(X)  where a is a constant and X is a random variable we calculate the variance of points.

Var(POINTS) =  4 * Var(Y) + 9 * Var(Z)

We know the expected value of the binomial random variable is np and the variance of the binomial random variable is np(1-p)

Using these formulas we can calculate the expected number of points and the variance of the number of points.  The calculations are laid out in the tables below.

 Expected Points from Two Basketball Teams Points Per make 2 3 n 15 5 p 0.5 0.35 Expected Points 15 5.25 20.25 Points Per make 2 3 n 10 10 p 0.5 0.35 Expected Points 10 10.5 20.50

Observation:  Expected points are identical in these two situations.

 Variance and Standard Deviation of Points Made by Two Basketball Teams Points Per make 2 3 n 15 5 p 0.5 0.35 1-p 0.5 0.65 Var 15 10.2375 25.24 STD 5.02 Points Per make 2 3 n 10 10 p 0.5 0.35 1-p 0.5 0.65 Var 10 20.475 30.48 STD 5.52

Observation:   The standard deviation of the team that relies on the 3-point shot is 9.9% higher than the standard deviation for the team that relies on the three point shot.    The variance for the three-point shot team is 21% higher than the variance for the team that depends more on the two point shot.

List of concepts used in this post:  This post teaches a number of statistical concepts including expected value and variance of the binomial distribution, the expected value and variance of the product of a constant and a random variable and the expected value and variance of the sum of two independent binomial random variables.

Outcomes for a team that uses both a two-point and three point shot can also be modeled with the multinomial distribution.  A future post will teach the multinomial distribution with a basketball example.

Other posts teaching probability through basketball can be found at the following post.