This post calculates the expected number and variance of
made free throws assuming the shooter is in the single bonus and assuming the shooter
is in the double bonus.
Question: In a previous post, we calculated the
outcomes and the probability of outcomes for a basketball player shooting fouls
in the single bonus and in the double bonus.
Go here if needed for a review of that calculation.
http://www.dailymathproblem.com/2019/05/freethrowoutcomesinsingleand.html
The chart below lists the probability of made free throw outcomes for two scenarios – one for a 60 percent free throw shooter and the other for an 80 percent free throw shooter under both the single and double bonus.
http://www.dailymathproblem.com/2019/05/freethrowoutcomesinsingleand.html
The chart below lists the probability of made free throw outcomes for two scenarios – one for a 60 percent free throw shooter and the other for an 80 percent free throw shooter under both the single and double bonus.
Probability
of making a free throw is 0.6


Single
Bonus

Double
Bonus


0
Makes

0.4

0.16

!
Makes

0.24

0.48

2
Makes

0.36

0.36

Total

1

1

Probability
of making a free throw is 0.8


Single
Bonus

Double
Bonus


0
Makes

0.2

0.04

!
Makes

0.16

0.32

2
Makes

0.64

0.64

Total

1

1

Find the expected number of
made free throws and the variance of made free throws for these two scenarios
for both freethrow success probabilities.
Discuss the implications of
these results.
Answer:
The expected value is calculated
by taking the SUMPRODUCT of the outcome vector with the probability vector.
Here are the expected value
calculations:
Expected
Value Calculation p=0.6


Single
Bonus

Double
Bonus


0

0.4

0.16

1

0.24

0.48

2

0.36

0.36

Expected
Value

0.96

1.2

Expected
Value Calculation p=0.8


Single
Bonus

Double
Bonus


0

0.2

0.04

1

0.16

0.32

2

0.64

0.64

Expected
Value

1.44

1.6

· Implication
observation: Note the improvement in expected value is 50
percent in the single bonus compared to 33.3% in the double bonus.
The variance calculation is
obtained by taking the SUMPRODUCT of the squared deviation between the outcome
(made free throws) and the expected outcome or expected free throws made.
Here are the variance
calculations for the single bonus situation for free throw success probability
of 0.6 and 0.8.
Variance
for Single Bonus Outcomes p=0.6


O

p

E(O)

(OE(0))2

0

0.4

0.96

0.9216

1

0.24

0.96

0.0016

2

0.36

0.96

1.0816

Variance

0.7584


Variance
for Single Bonus Situation =0.8


O

p

E(O)

(OE(0))2

0

0.2

1.44

2.0736

1

0.16

1.44

0.1936

2

0.64

1.44

0.3136

Variance

0.6464

O is outcome 0, 1 or 2 makes. P is probability for each
outcome. E(O) is expected number of free
throws made. (OE(O))^{2} is
difference between outcome (number of makes and expected outcome or expected
number of makes squared. Variance is SUMPRODUCT
of probability of each outcome and (OE(O))^{2}
for each outcome.
Here are the variance calculations
for the double bonus situations for both freethrow make probabilities.
Variance
for Double Bonus with p=0.6


O

p

E(O)

(OE(0))2

0

0.4

1.2

1.44

1

0.24

1.2

0.04

2

0.36

1.2

0.64

Variance

0.816


Variance
for Double Bonus p=0.8


O

p

E(O)

(OE(0))2

0

0.2

1.6

2.56

1

0.16

1.6

0.36

2

0.64

1.6

0.16

Variance

0.672

O is outcome 0, 1 or 2 makes. P is probability for each
outcome. E(O) is expected number of free
throws made. (OE(O))^{2} is
difference between outcome (number of makes and expected outcome or expected
number of makes squared. Variance is SUMPRODUCT
of probability of each outcome and (OE(O))^{2}
for each outcome.
· Implication Observation: The increase
in free throw success probability results in a decrease in the variance of the
number of free throws by 14.7 % in the single bonus situation and by 17.6 % in
the double bonus situation.
No comments:
Post a Comment