The Role of Three
Point Shots
Question: Consider two basketball teams. The first team takes 15 twopoint shots and
5 threepoint shots. The second team
takes 10 twopoint shots and 10 three point shots. On average, both teams make 50% of their
twopoint shots and 35% of their twopoint 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 twopoint
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(n_{2},0.5) and
Z=B(n_{3},0.35) where n_{2 }is the number of twopoint shots
made and n_{3} 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 a^{2 }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(1p)
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


1p

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


1p

0.5

0.65


Var

10

20.475

30.48

STD

5.52

Observation: The standard deviation of the team that
relies on the 3point shot is 9.9% higher than the standard deviation for the
team that relies on the three point shot.
The variance for the threepoint 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 twopoint 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.
No comments:
Post a Comment