## Monday, April 13, 2015

### Bonus free throws -- post number one.

Analysis of bonus free throws

This post considers math associated with free throw shots when a team is in the bonus situation.   A team gets bonus free throws when the other team makes a lot of fouls.  I believe the team gets in the bonus when the other team makes 5 fouls.

A foul that is not in the act of shooting may lead to one free throw.   When few fouls have been committed there may not be any shots.   The player and team committing the foul are assigned with a foul but the fouled player does not get to immediately shoot.  Once a few fouls gests committed the team that is fouled gets to shoot.

When more fouls have been committed a team get to take bonus shots.  If a team is in the bonus the player will get to take a second shot if he makes the first shot.   Miss the first shot and there is no second shot.

Questions:

What is the probability that the player will make 0, 1, or 2 free throws when a player is fouled and the team is in a bonus situation?

What is the expected number and variance of shots made?

What is the probability that the player makes at least one free throw in the two situations?

Evaluate this situation for a player with a 60% likelihood of making a free throw and a 80% likelihood of making a free throw?

Analysis:

The probability of making 0 shots is (1-p).

The probability of one shot is (1-p) x p.  Here (1-p) is the probability of missing the first and p is the probability of hitting the second.

The probability of making two shots is p x p.

I evaluate the probability of making 0, 1, or 2 shots, the expected number of makes and the variance of makes in the following table.

 Calculation For Bonus Situation # Makes Prob (Make=k) p=0.6 p=0.8 0 (1-p) 0.4 0.2 1 (1-p) x p 0.24 0.16 2 p x p 0.36 0.64 E(Makes) 0.96 1.44 E(Makes^2) 1.68 2.72 Var(Makes) 0.7584 0.6464

Note:  P(makes=0) + P(makes=1) + P(makes=2)    =   1   Phew!

At least one make means one or two makes.

At p =-0.6 the probability of at least one make is 0.60.

At p=0.8 the probability of at least one make is 0.80.

An increase in the probability of making a free throw from 0.6 to 0.80 is a 33% increase.

This translates into a 50% increase in the expected number of free throws made in a bonus situation..  (A change from 0.96 to 1.44 is a 50% change.)

Also the variance falls by 14.8 ( a change from 0.7584 to 0.6454).

Notes:

I am planning to write more posts on bonus free throw situations.  They will be available soon.

Readers who liked this post may also want to look at one of my previous post on three-point shots and risk.

I have written a number of reports on various topics on Kindle.   You make like my report on the 2014 election.