Negative Binomial Distribution
2026-02-28 17:15 Diff
https://brightchamps.com/en-us/math/algebra/negative-binomial-distribution

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Last updated on October 22, 2025

The negative binomial distribution tells us how many trials it takes until we reach a certain number of successes, focusing on the trial in which the final success occurs. In this article, we’ll explore this concept in more detail.

What is Negative Binomial Distribution?

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The negative binomial distribution models the number of trials needed to achieve r successes, assuming each trial has the same probability of success, θ. Here, successes are fixed and trials are variable. If you repeat independent trials until the rᵗʰ success occurs, the probability that this happens after x failures (i.e., on the x + rth trial) is given by: 
 

\(f(x) = \left( \frac{x + r - 1}{r - 1} \right) p r^x, \quad x = 0, 1, 2, \dots \)

Where:
x = number of failures before the r-th success
r = total number of successes we want to observe
p = probability of success on a single trial
q = probability of failure on a single trial = 1 − p
 

Characteristics of the Negative Binomial Distribution

The objective of the negative binomial distribution is to help us figure out how many trials it might take to get a certain number of successes when each trial has the same chance of success. It depends mainly on two things: 

  • r - the required number of successes, which should be a whole number.
  • p - the probability of success in each trial (between 0 and 1).
  • These two values shape how the distribution looks and how it behaves.

For example:
Let’s say a basketball player wants to make 4 successful free throws. 

  • So, r = 4 
  • Let’s say their chance of making a shot is p = 0.3 (30%)

This setup tells us how many total attempts we might expect before the player gets those 4 makes. 

  • If we increase r, the player will need more tries, and the results can vary more. 
  • If we increase p, the player will probably need fewer attempts, and the number of trials becomes more predictable. 

What are the Properties of Negative Binomial Distribution?

The properties of the negative binomial distribution are as follows:

  • The experiment consists of independent and identical trials, repeated until a fixed number of successes (r) is observed. 
  • Each trial results in either a success or a failure. 
  • The probability of success (p) stays the same in every trial. The probability of failure is q = 1 - p. 
  • The outcome of one trial does not influence the outcome of another. 
  • The experiment continues until a specific number of successes (r) is reached. The value of r is chosen in advance. 
  • The total number of trials varies and is equal to x + r, where x is the number of failures before the final (r-th) success.

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Probability Density Function (PDF) of Negative Binomial Distribution

The probability density function of the negative binomial distribution gives the probability of observing a specific number of failures before reaching a fixed number of successes. It is given by: 

\(P(X = x) = \left( \frac{x + r - 1}{r - 1} \right) p^r (1 - p)^x, \quad x = 0, 1, 2, \dots \)

Where,
x = number of failures before rth success
r = required number of successes
p = probability of success in each trial 
(1-p) = q = probability of failure, and 
The binomial coefficient shows how many different ways the failures and successes can be ordered before the final success occurs. 
It describes the probability that the rth success occurs exactly on trial x + r.
 

What are Mean and Variance of Negative Binomial Distribution

The mean and variance tell us what to expect from the number of trials needed to reach a set number of successes.

  • Mean: 
    Mean = rp. It represents the average number of trials needed to get r successes, where p is the probability of success in each trial. 
  • Variance:
    Variance = r(1 - p)p2. It shows how much the number of trials might vary around the average.

Example:
If a player has a 40% chance of scoring on each attempt and wants to score 3 times, then:

  • Mean = 3/0.4 = 7.5 trials on average (p = 0.4 and r = 3)
  • Variance = 3(1 - 0.4) / (0.4)2 = 3(0.6) / 0.16 = 11.25 
     

Common Mistakes and How to Avoid Them in Negative Binomial Distribution

While working with the negative binomial distribution, students may confuse it with the binomial distribution, among other common mistakes. In this section, we will identify these mistakes and learn how to avoid repeating them.

Real-life Applications of Negative Binomial Distribution

Negative binomial distribution has a wide range of applications in various fields like hospitality, R&D, and marketing. Here, we’ve mentioned a few applications.

  • Customer service/ sales calls: A salesperson may want to know how many phone calls they need to make before successfully closing 5 deals, assuming each call has a fixed chance of success.
  • Determining the number of patients in medical trials: In clinical research, the distribution can model how many patients need to be treated until a certain number of successful outcomes (e.g., patients cured) are observed, with each treatment having a fixed success probability.
  • Quality Control in Manufacturing: Factories can use this method to estimate the number of items that need to be inspected before finding a certain number of defective pieces, especially if the defects occur randomly.
  • Student testing in educational institutions: A teacher can find out how many attempts a student will need to pass a certain number of modules, given a certain chance of passing each one.
  • Key performance indicators in digital marketing:Negative binomial distribution can help predict how much website traffic or ad clicks are required before a set number of purchases occur.

Problem 1

A basketball player makes a successful shot with a probability of 0.6. What is the probability that she makes her 3rd successful shot on the 5th attempt?

Okay, lets begin

0.20736

Explanation

We will use the negative binomial distribution, which gives the probability that the r-th success happens on the x-th trial.
Here:
r = 3 (we want the 3rd success)
x = 5 (shot is successful on the 5th attempt)
p = 0.6 (probability of success)
q = 1 - p = 0.4 (probability of failure)
The negative binomial probability formula is:

\(P(X = x) = \left( \frac{x - 1}{r - 1} \right) p^r q^{x - r} \)

Substituting the values of x, r, p, and q in the formula, we get: 

\(P(X = 5) = \binom{4}{2} (0.6)^3 (0.4)^2 = 6 \cdot 0.216 \cdot 0.16 = 0.20736 \)

Well explained 👍

Problem 2

A fair coin (p = 0.5) is tossed until we get 3 heads. What is the probability that this takes exactly 5 tosses?

Okay, lets begin

0.1875

Explanation

We’re tossing a fair coin and want the third head to happen on the 5th toss. That means: 

  • There must be 2 heads somewhere in the first 4 tosses
  • And the 5th toss must be a head (the final, 3rd success) 

So, we’re counting the number of failures (tails) before we reach the 3rd success (head).

  • r = 3 (we want 3 heads)
  • p = 0.5 (fair toss)
  • x = 2 (we need 2 tails before the 3rd head)

\(P(X = 2) = \left( \frac{2 + 3 - 1}{3 - 1} \right) (0.5)^3 (0.5)^2 = \binom{4}{2} \cdot 0.125 \cdot 0.25 = 6 \cdot 0.03125 = 0.1875 \)

Well explained 👍

Problem 3

A machine produces 10% defective items. What is the probability that the 2nd non-defective (success) item is found on the 5th trial?

Okay, lets begin

 0.00324

Explanation

We’re looking for the probability that the 2nd non-defective item comes up on the 5th trial. That means:
The first 4 items must include exactly 1 non-defective and 3 defective items
The 5th item must be non-defective
We’re using the negative binomial distribution, where:
r = 2 
p = 0.9
x = 3
Substituting the values in the formula, we get:
\(P(X = 3) = \left( \frac{3 + 2 - 1}{2 - 1} \right) (0.1)^3 (0.9)^2 = \binom{4}{1} \cdot 0.001 \cdot 0.81 = 4 \cdot 0.00081 = 0.00324 \)

Well explained 👍

Problem 4

A customer makes a purchase 40% of the time. What’s the probability that the 3rd purchase happens on the 7th visit?

Okay, lets begin

15⋅0.0082944 ≈ 0.1244

Explanation

p = 0.4, r = 3, total trials = 7 ⇒ x = 4
\(P(X = 4) = \left( \frac{4 + 3 - 1}{2} \right) \cdot (0.6)^4 \cdot (0.4)^3 = \binom{6}{2} \cdot 0.1296 \cdot 0.064 \)

Well explained 👍

Problem 5

In a drug trial, each test has a 20% chance of success. What is the probability that the 5th success occurs on the 12th trial?

Okay, lets begin

≈0.0221

Explanation

We want the probability that the 5th success happens exactly on the 12th trial. That means:
In the first 11 trials, there must be 4 successes and 7 failures (in any order)
The 12th trial must be a success 
To solve this, we use the following formula:

\(P(X = 7) = \left( \frac{7 + 5 - 1}{4} \right) \cdot (0.8)^7 \cdot (0.2)^5 = \binom{11}{4} \cdot 0.2097152 \cdot 0.00032 \)

so: 330 ⋅ 0.2097152 ⋅ 0.00032 ≈ 0.0221
 

Well explained 👍

FAQs on Negative Binomial Distribution

1.What is the negative binomial distribution used for?

It is used to model the number of trials needed to achieve a fixed number of successes in independent trials.

2.How is the negative binomial distribution different from the binomial distribution?

The difference between the negative binomial distribution and the binomial distribution is given in the table below:

Binomial distribution

Negative binomial distribution

The number of trials (n) is fixed

The number of successes (r) is fixed

The number of successes achieved in those trials is not fixed

The number of trials needed to get the r-th success is not fixed

It tells us how many successes happen within a fixed number of trials

It tells us how many failures happen before reaching a fixed number of successes

The binomial distribution is calculated by using the formula:

The formula to find the negative binomial distribution is:

3.What are the key assumptions of the negative binomial distribution?

The key assumptions are:

  1. Each trial results in either a success or a failure — no other outcome is possible. 
  2. The chance of success remains constant in every trial.
  3. Trials are independent.
  4. The process continues until a set number of successes has been reached.
     

4. What Is The Formula For the Negative Binomial Distribution?

The formula for the negative binomial distribution is:

\(f(x) = \left( \frac{x + r - 1}{r - 1} \right) p^r q^x \)

x = 0, 1, 2,...
Where p = success probability, q=1−p

5.Can the negative binomial distribution be used when data shows more variability than expected?

Yes, the negative binomial distribution is well-suited for overdispersed data, where the variance is greater than the mean.

Jaskaran Singh Saluja

About the Author

Jaskaran Singh Saluja is a math wizard with nearly three years of experience as a math teacher. His expertise is in algebra, so he can make algebra classes interesting by turning tricky equations into simple puzzles.

Fun Fact

: He loves to play the quiz with kids through algebra to make kids love it.