**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.

Questions
and links below:

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.

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