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.
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