## Thursday, April 11, 2013

### Returns and risks from three-point and two-point shots

Question: A shooter takes 15 three-point shots.   The shots are independent and identically distributed. The likelihood the shot goes in is 0.300.   What is the expected number of points scored from these 15 shots?   What likelihood of a shot going in must a two-point shooter who takes 15 shots have in order to have the same number of expected points?

What is the standard deviation of points over the 15 shots for the three-point shooter?  What is the standard deviation for the two-point shooter?

Answer:  Each shot is a random variable X with probability going in p.

The expected number of points scored for the three-point shooter is

E(POINTS)=E(3X1 + 3X2 + …..3X15 )

E(POINTS)=15 x 0.03 x 3 = 13.5

For the two-point shooter to have 13.5 expected points in 15 shots the following must hold

13.5=15*2*p

or the shooting percentage must be p=13.5/30, which is 0.45.

The random variable for whether a person makes or misses a shot is binary  -- 1 if the shooter makes the shot, 0 if the shooter misses.  Point scored are 2 for the two-point shot and 3 for the 3-point shot.

The variance of points for one shot is

Var(Points per one shot)=SPM2 x P x (1-P)

where SPM is shot per make , p is the probability of the make, and (1-p) the probability of a miss.

Our shooters are each taking 15 independent shot.  The variance of independent events is additive.

Using these facts the variance of points from 15 independent shots for the three-point shooter is

Var(Points)=15 x 9 x 0.30 x 0.70 =28.35

The standard deviation of the square root of the variance is 5.32

The variance of points for the two-point shooter is

Var(Points)=15 x 4 x 0.45 x 0.55 = 14.85

The standard deviation for the two-point shooter is 3.85.

Comment:  The large variability in points is one reason why there are large lead swings in games where one or both teams rely on the three-point shot.     Both Louisville and Michigan relied a lot on the three point shot in the National Championship game.  Defense was probably the deciding factor in this game.

The extent to which points in a game are skewed is also important.
I hope to write more about points scored soon.

If you liked this post you may also like the post on hitting streaks and the likelihood of a sweep.

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