This post lists freethrow outcomes and the probability
of freethrow 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 1p 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 (1p)^{2} or the probability of taking two shots and missing
both.
The probability of making one
shot in the single bonus is p*(1p). 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*(1p)*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 p^{2} 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 1p + p*(1p) + p*p adds
to 1.
· Note also that for the double bonus (1p)^{2 }+ 2*p*(1p) + p^{2} simplifies to
1.0.
Calculations for the single bonus situation:
Single
Bonus Calculations


p

0.6


1p

0

0.4

p*(1p)

1

0.24

p^{2}

2

0.36

Total

1


p

0.7


1p

0

0.3

p*(1p)

1

0.21

p^{2}

2

0.49

Total

1


p

0.8


1p

0

0.2

p*(1p)

1

0.16

p^{2}

2

0.64

Total

1

Calculations for the double bonus situation:
Double
Bonus Calculations


p

0.6


(1p)^{2}

0

0.16

2*p*1p)

1

0.48

p^{2}

2

0.36

Total

1


p

0.7


(1p)^{2}

0

0.09

2*p*1p)

1

0.42

p^{2}

2

0.49

Total

1


p

0.8


(1p)^{2}

0

0.04

2*p*1p)

1

0.32

p^{2}

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.
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