## Friday, May 17, 2019

### Free throw makes under first and second bonus – expected value and variance

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/free-throw-outcomes-in-single-and.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 free-throw success probabilities.

Discuss the implications of these results.

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) (O-E(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) (O-E(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.  (O-E(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 (O-E(O))2  for each outcome.

Here are the variance calculations for the double bonus situations for both free-throw make probabilities.

 Variance for  Double Bonus with p=0.6 O p E(O) (O-E(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) (O-E(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.  (O-E(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 (O-E(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.