## Friday, April 17, 2015

### The statistics of bonus free throws

Questions on Bonus Free Throws

In basketball a team gets bonus free throws when the other team commits a large number of fouls.   The first shot is crucial when a team is in the bonus situation because the player can take the second shot only if he makes the first shot.

Bonus free throws are a bit more complicated than regular free throws.   One regular free throw can be described by a binary variable.   The sum of the binary variables is binomially distributed.     A bonus free throw has three out comes 0, 1, or 2 makes.   The sum of two bonus situations has four outcomes 0, 1, 2, 3, or 4 makes.  The sum of four independent regular free throw shots has outcomes of 1 to 4 makes but expected makes are higher for the person who takes four regular shots than for the person who takes 2 bonus situations.

I’ve written three posts on the statistics explaining bonus free throws.   The posts utilize concepts pertaining to probability distribution functions, independent events, expected value and variance.  I hope they are useful to you.

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?

Consider two basketball situations.   Situation one is a player with four guaranteed free throws.  Situation two is a player with two bonus free throw situations.  (Recall that under the bonus the player needs to make the first shot to get the second shot.)  What are the possible outcomes in terms of shots made and the likelihood of made shots in the two situations?   Evaluate these two situations under the assumption that each free throw is independent and identically distributed.   Evaluate the situations for two free throw shooters, one with a sixty percent likelihood of making each shot and the other with a eighty percent success rate.

What is the expected number and variance of free throw makes for a person with 5 bonus free throw opportunities and for a person with 10 regular (not in bonus)  free throw opportunities?    Provide calculations for a 60 percent free throw shooter and a 90 percent free throw shooter?

The people who like this post might also find my book Statistical Applications of Baseball useful.