Negative Binomial Distribution
Negative binomial distribution talks about the final success which can be obtained, after a sequence of successes in the preceding trials. Negative binomial distribution refers to the r^{th} success which has been preceded by n  1 trials, containing r  1 success.
Let us learn more about the negative binomial distribution, formula, and properties of negative binomial distribution, with the help of examples, FAQs.
What Is Negative Binomial Distribution?
The negative binomial distribution is the distribution of the number of trialn needed to get r^{th} successes. The negative binomial distribution helps in finding r success in x trials. Here we aim to find the specific success event, in combination with the previous needed successes. In negative binomial distribution, the number of trials and the probability of success in each trial are defined clearly. Here we consider the n + r trials needed to get r successes.
Negative Binomial Distribution: f(x) = \(^{n + r  1}C_{r  1}.P^r.q^n\)
A binomial experiment is an experiment consisting of a fixed number of independent Bernoulli trials. Each Bernoulli trial is an independent trial and has two possible outcomes, occurrence or nonoccurrence (success or failure), and each trial has the same probability of occurrence of successes. The negative binomial distribution talks about the distribution of the number of trials needed to get the defined number of successes. The negative binomial distribution is almost the same as a binomial distribution with one difference: In a binomial distribution we have a fixed number of trials, but in negative binomial distribution we have a fixed number of successes.
A random variable X is supposed to follow a negative binomial distribution if its probability mass function is given by:
f(x) = (n + r  1)C(r  1) P^{r}q^{x}, where x = 0, 1, 2, ....., and p + q = 1.
Here we consider a binomial sequence of trials with the probability of success as p and the probability of failure as q.
Let f(x) be the probability defining the negative binomial distribution, where (n + r) trials are required to produce r successes. Here in (n + r  1) trials we get (r  1) successes, and the next (n + r) is a success.
Then f(x) = (n + r  1)C(r  1) P^{r1}q^{n1}.p
f(x) = (n + r  1)C(r  1) P^{r}q^{n}
Additional Points of Negative Binomial Distribution
The following are the three important points referring to the negative binomial distribution.
 The mean of the negative binomial distribution is E(X) = rq/P
 The variance of the negative binomial distribution is V(X)= rq/p^{2}
 Here the mean is always greater than the variance. Mean > Variance.
Negative binomial distribution takes an account of all the successes which happen one step before the actual success event, which is further multiplied by the actual success event. Since it takes an account of all the successes one step before the actual success event, it is referred to as a negative binomial distribution. Negative binomial distribution refers to the r^{th} success which has been preceded by n  1 trial, containing r  1 success.
A negative binomial distribution is also called a pascal distribution.
Examples Of Negative Binomial Distribution
The following quick examples help in a better understanding of the concept of the negative binomial distribution.
 If we flip a coin a fixed number of times and count the number of times the coin turns out heads is a binomial distribution. If we continue flipping the coin until it has turned a particular number of heads say the third headon flipping 5 times, then this is a case of the negative binomial distribution.
 For a situation involving three glasses to be hit with 7 balls, the probability of hitting the third glass successfully with the seventh ball can be obtained with the help of negative binomial distribution.
 In a class, if there is a rumor that there is a math test, and the fifth is the second person to believe the rumor, then the probability of this fifth person to be the second person to believe the rumor can be computed using the negative binomial distribution.
Properties Of Negative Binomial Distribution
A negative binomial distribution is a distribution that has the following properties.
 The negative binomial distribution has a total of n number of trials.
 Each trial has two outcomes, and one of them is referred to as success and the other as a failure.
 The probability of success or failure is the same across each of these trials.
 The probability of success is denoted by p, and the probability of failure is defined as q, and each of these is the same in every trial.
 The sum of the probability of success and failure is equal to 1. p + q = 1.
 Each of these trials is independent. The outcome of one trial does not affect the outcome of other trials.
 The experiment is continued until r success is obtained, and r is defined in advance.
 The experiment consists of x + r repeated trials, where r is the required number of successes.
Related Topics
The following topics help in a better understanding of negative binomial distribution.
Examples on Negative Binomial Distribution

Example 1: Jim is writing an exam with multiplechoice questions, and his probability of attempting the question with the right answer is 60%. What is the probability that Jim gives the third correct answer for the fifth attempted question?
Solution:
Probability of success P(s) = 60% = 0.6, Probability of failure P(f) = 40% = 0.4.
It is given that Jim gives the third correct answer for the fifth attempted question.
Here we can use the concept of the negative binomial distribution to find the third correct answer for the fifth attempted question.
Here we have x = 5, r = 3, P = 0.6, q = 0.4
The formula for negative binomial distribution is B(x, r, P) = (x  1)C(r  1)P^{r}.Q^{x  r}
= (5  1)C(3  1). (0.6)^{3}.(0.4)^{5  3}
= 4C2.(0.6)^{3}.(0.4)^{2}
= 6.(0.216)(0.16)
= (0.1296)(0.16)
= 0.020736
Therefore the probability of Jim giving the third correct answer for his fifth attempted question is 0.02.

Example 2: The probability of Ron going on time to school is 80%. What is the probability that Ron goes on time for the eighth time for the first ten days of school?
Solution:
Probability of Success P(S) = 80% = 0.8, Probability of Failure P(f) = 20% = 0.2
It is given that Ron goes on time for the eighttime for the first ten days of school.
Here also we can use the negative binomial distribution to find the eighth day when he goes on time to school, for the first ten days of school.
Here we have x = 10, r = 8, P = 0.8, q = 0.2
The formula for negative binomial distribution is B(x, r, P) = (x  1)C(r  1)P^{r}.Q^{x  r}
= (10  1)C(8  1). (0.8)^{8}.(0.42^{10  8 }
= 9C7.(0.8)^{8}.(0.2)^{2}
= 36.(0.281)(0.04)
= 36(0.01124)
= 0.40464
Therefore the probability of Ron going on time for the first ten days is 0.4.
FAQs on Negative Binomial Distribution
What Is Negative Binomial Distribution?
The negative binomial distribution is the distribution of the number of trials needed to get r^{th} successes. The negative binomial distribution helps in finding r success in x trials. Here we aim to find the specific success event, in combination with the previous needed successes. The formula for negative binomial distribution is f(x) = \(^{n + r  1}C_{r  1}.P^r.q^x\).
What Is The Formula For Negative Binomial Distribution?
The formula for negative binomial distribution is f(x) = \(^{n + r  1}C_{r  1}.P^r.q^x\). Here n + r is the total number of trials, and r refers to the r^{th} success. Also, p refers to the probability of success, and q refers to the probability of failure, and p + q = 1.
Why Is This Called Negative Binomial Distribution?
Negative binomial distribution takes an account of all the successes which happen one step before the actual success event, which is further multiplied by the actual success event. Since it takes an account of all the successes one step before the actual success event, it is referred to as a negative binomial distribution. Negative binomial distribution refers to the r^{th} success which has been preceded by n  1 trial, containing r  1 success.
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