Thursday, May 16, 2019

Free Throw Outcomes in Single and Double Bonus


This post lists free-throw outcomes and the probability of free-throw outcomes for a basketball player shooting fouls under the single bonus and under the double bonus.

Information on Single and Double Bonus in Basketball:

·      When the opposing team commits seven fouls the fouled team receives a single bonus free throw on common fouls.   When in the single bonus, the fouled team gets an extra free throw if and only if the player makes the first free throw on a common foul.

·      When the opposing team commits ten fouls the fouled team is in the double bonus and gets two free throws on a common foul regardless of whether the person makes  the first free throw attempts.

Question:  What are the possible outcomes and the probability of each outcome defined by the number of made free throws for a basketball player shooting fouls when in the single bonus and when in the double bonus?  Calculate the likelihood of each outcome when the probability the shooter makes a free throw is 0.6, 0.7, and 0.8.

Outcome and probability of outcomes for single and double bonus situations:

In both situations there are three outcomes 0 makes, 1 make, and 2 makes.

The probability of zero makes in the single bonus is 1-p where p is the probability of making a free throw.   Miss the first shot and shooter is done.  


The probability of zero makes in the double bonus is (1-p)2 or the probability of taking two shots and missing both.


The probability of making one shot in the single bonus is p*(1-p).  This happens by the player sinking the first shot and missing the second.


The probability of making one shot in the double bonus is p*(1-p)*2.    The player can make one shot by making the first shot and missing the second or missing the first shot and making the second.

The probability of the player making both shots is p2 in both the first and second bonus.


Notes:  The sum of the probabilities of all outcomes in the sample space must sum to 1.

·      Note for the single bonus 1-p + p*(1-p)  + p*p  adds to 1.  
·      Note also that for the double bonus  (1-p)2 + 2*p*(1-p) + p2  simplifies to 1.0. 

Calculations for the single bonus situation:

Single Bonus Calculations
p
0.6
1-p
0
0.4
p*(1-p)
1
0.24
p2
2
0.36
Total
1
p
0.7
1-p
0
0.3
p*(1-p)
1
0.21
p2
2
0.49
Total
1
p
0.8
1-p
0
0.2
p*(1-p)
1
0.16
p2
2
0.64
Total
1



Calculations for the double bonus situation:

Double Bonus Calculations
p
0.6
(1-p)2
0
0.16
2*p*1-p)
1
0.48
p2
2
0.36
Total
1
p
0.7
(1-p)2
0
0.09
2*p*1-p)
1
0.42
p2
2
0.49
Total
1
p
0.8
(1-p)2
0
0.04
2*p*1-p)
1
0.32
p2
2
0.64
Total
1



Discussion:

The probability of making both free throws is identical for both the single and double bonus.

The probability of making no free throws or one free throw is higher in double bonus than single bonus.

A solid free throw shooter with the 0.8 probability has a 4 percent chance of zero free throws when in the double bonus compared to a 20 percent chance in the single bonus.










No comments:

Post a Comment