Negative Binomial Distribution
In this lesson, we cover the negative binomial distribution and the
geometric distribution. As we will see, the geometric distribution
is a special case of the negative binomial distribution.
Negative Binomial Experiment
A negative binomial experiment is a
statistical experiment that has the following properties:
 The experiment continues until r successes are observed, where r
is specified in advance.
Consider the following statistical experiment. You flip a coin repeatedly and count
the number of times the coin lands on heads. You continue flipping the coin until
it has landed 5 times on heads. This is a negative binomial experiment
because:
 The experiment continues until a fixed number of successes have occurred;
in this case, 5 heads.
Notation
The following notation is helpful, when we talk about negative binomial probability.
 n!: The factorial of n (also known as n factorial).
Negative Binomial Distribution
A negative binomial random variable is the number X of
repeated trials to produce r successes in a negative binomial
experiment. The
probability distribution
of a negative binomial random variable is called a negative binomial
distribution. The negative binomial distribution is also known
as the Pascal distribution.
Suppose we flip a coin repeatedly and count the number of heads (successes).
If we continue flipping the coin until it has landed 2 times on heads, we
are conducting a negative binomial experiment. The negative
binomial random variable is the number of coin flips required to achieve
2 heads. In this example, the number of coin flips is a random variable
that can take on any integer value between 2 and
plus infinity. The negative binomial probability distribution for
this example is presented below.
Number of coin flips

Probability 
2

0.25 
3

0.25 
4

0.1875 
5

0.125 
6

0.078125 
7 or more

0.109375 
Negative Binomial Probability
The negative binomial probability refers to the
probability that a negative binomial experiment results in
r  1 successes after trial x  1 and
r successes after trial x. For example,
in the above table, we see that the negative binomial probability of
getting the second head on the sixth flip of the coin is 0.078125.
Given x, r, and P, we can compute the negative binomial probability
based on the following formula:
Negative Binomial Formula. Suppose a negative binomial
experiment consists of x trials and results in r successes.
If the probability of success on an individual trial is P,
then the negative binomial probability is:
b*(x; r, P) = _{x1}C_{r1} * P^{r} * (1  P)^{x  r}
b*(x; r, P) = { (x1)! / [ (r1)!(xr)!] } * P^{r} * (1  P)^{x  r}
The Mean of the Negative Binomial Distribution
If we define the mean of the negative binomial distribution
as the average number of trials required to produce r successes, then the mean
is equal to:
μ = r / P
where μ is the mean number of trials, r is the number of successes,
and P is the probability of a success on any given trial.
Alternative Views of the Negative Binomial Distribution
As if statistics weren't challenging enough, the above definition is not the
only definition for the negative binomial distribution.
Two common alternative definitions are:
 The negative binomial random variable is R, the number of successes before the
binomial experiment results in k failures.
The mean of R is:
μ_{R} = kP/Q
 The negative binomial random variable is K, the number of failures before the
binomial experiment results in r successes.
The mean of K is:
μ_{K} = rQ/P
The moral: If someone talks about a negative binomial
distribution, find out how they are defining the negative binomial random
variable.
On this website, when we refer to the negative binomial distribution,
we are talking about the definition presented earlier. That is, we are
defining the negative binomial random variable as X, the total number of
trials required for the binomial experiment to produce r successes.
Geometric Distribution
The geometric distribution is a special case of the
negative binomial distribution. It deals with the number of trials
required for a single success. Thus, the geometric distribution is
negative binomial distribution where the number of successes (r)
is equal to 1.
An example of a geometric distribution would be tossing a coin until
it lands on heads. We might ask: What is the probability that the
first head occurs on the third flip? That probability is referred
to as a geometric probability and is denoted by
g(x; P). The formula for geometric probability is
given below.
Geometric Probability Formula. Suppose a negative binomial
experiment consists of x trials and results in one success.
If the probability of success on an individual trial is P,
then the geometric probability is:
g(x; P) = P * Q^{x  1}
Test Your Understanding
The problems below show how to apply your newfound knowledge of the
negative binomial distribution (see Example 1) and the geometric
distribution (see Example 2).
Negative Binomial Calculator
As you may have noticed, the negative binomial formula requires many timeconsuming
computations. The Negative Binomial Calculator can do this work for you  quickly,
easily, and errorfree. Use the Negative Binomial Calculator to compute negative binomial
probabilities. The
calculator is free. It can found in the Stat Trek
main menu under the Stat Tools tab. Or you can tap the button below.
Negative Binomial Calculator
Example 1
Bob is a high school basketball player. He is a 70% free throw shooter.
That means his probability of making a free throw is 0.70. During the
season, what is the probability that Bob makes his third free throw
on his fifth shot?
Solution: This is an example of a negative binomial experiment.
The probability of success (P) is 0.70,
the number of trials (x) is 5,
and the number of successes (r) is 3.
To solve this problem, we enter these values into the negative binomial
formula.
b*(x; r, P) =
_{x1}C_{r1} * P^{r} * Q^{x  r}
b*(5; 3, 0.7) =
_{4}C_{2} * 0.7^{3} * 0.3^{2}
b*(5; 3, 0.7) = 6 * 0.343 * 0.09 = 0.18522
Thus, the probability that Bob will make his third successful free
throw on his fifth shot is 0.18522.
Example 2
Let's reconsider the above problem from Example 1. This time,
we'll ask a slightly different question: What is the probability that
Bob makes his first free throw on his fifth shot?
Solution: This is an example of a geometric distribution, which is
a special case of a negative binomial distribution. Therefore, this
problem can be solved using the negative binomial formula or the
geometric formula. We demonstrate each approach below, beginning with
the negative binomial formula.
The probability of success (P) is 0.70,
the number of trials (x) is 5,
and the number of successes (r) is 1.
We enter these values into the negative binomial formula.
b*(x; r, P) =
_{x1}C_{r1} * P^{r} * Q^{x  r}
b*(5; 1, 0.7) =
_{4}C_{0} * 0.7^{1} * 0.3^{4}
b*(5; 3, 0.7) = 0.00567
Now, we demonstate a solution based on the geometric formula.
g(x; P) = P * Q^{x  1}
g(5; 0.7) = 0.7 * 0.3^{4} = 0.00567
Notice that each approach yields the same answer.