---
title: "Negative Binomial Distribution | Intro to Statistics"
description: "Negative binomial distribution models the number of trials or failures needed to get a set number of successes in Intro to Statistics and beyond."
canonical: "https://fiveable.me/college-intro-stats/key-terms/negative-binomial-distribution"
type: "key-term"
subject: "Intro to Statistics"
unit: "Unit 4"
---

# Negative Binomial Distribution | Intro to Statistics

## Definition

The negative binomial distribution is a discrete distribution for counting how many Bernoulli trials or failures happen before you reach a fixed number of successes. In Intro to Statistics, it extends the geometric distribution past the first success.

## What It Is

The negative binomial distribution is the Intro to Statistics model you use when you keep running independent Bernoulli trials until you reach a set number of successes. Instead of asking only about the first success, it tracks how long it takes to get the rth success when each trial has the same success probability p.

That makes it a natural next step after the geometric distribution. Geometric distribution asks, “How many trials until the first success?” Negative binomial asks the same kind of question, but for the second, third, or nth success. The setup still needs independent trials and a constant probability of success, so you are usually thinking about coin flips, repeated quality checks, survey responses, or other yes-or-no processes.

There are two common ways to define the random variable. Some classes let X be the number of trials needed to get r successes. Others let X be the number of failures before the rth success. The formula changes depending on which version your course uses, so the first thing to check is what X is counting. If your instructor or textbook says “before the rth success,” then failures are being counted directly.

The probability pattern comes from counting how many trial orders can produce the same result. For example, to get the rth success on the xth trial, the last trial must be a success, and the first x minus 1 trials must contain exactly r minus 1 successes. That is why combinations show up in the PMF. You are not just multiplying probabilities, you are also counting the different arrangements that lead to the same outcome.

A compact example makes this easier to see. Suppose you want the probability that the 3rd success happens on the 5th trial. Then the 5th trial must be a success, and among the first 4 trials you need exactly 2 successes and 2 failures. The negative binomial model handles that kind of “keep going until a target number of successes” question in one distribution.

One common mistake is confusing this with the binomial distribution. Binomial fixes the number of trials and counts successes. Negative binomial fixes the number of successes and counts how many trials or failures it took to get there.

## Why It Matters

Negative binomial distribution shows up whenever Intro to Statistics moves from fixed-length experiments to “keep trying until” experiments. That shift matters because the sample space changes. With a binomial model, you know the number of trials ahead of time. With a negative binomial model, the stopping point depends on when the success target is reached.

This is also where students practice reading a probability question carefully. If the problem says “the third successful sale,” “the fifth passed inspection,” or “until two customers say yes,” you are probably not in binomial land anymore. You need to identify what is being counted, what counts as success, and whether the trials are independent with the same p each time.

It also connects directly to the geometric distribution unit. Geometric is the special case where r = 1, so negative binomial gives you a bigger version of the same idea. Once you see that relationship, it becomes easier to tell whether a problem is asking about the first success only or about a later success after several rounds.

In problem sets, this distribution helps you set up probability calculations instead of guessing formulas. In class discussion or quizzes, it often shows up as a word problem where the main challenge is choosing the right model. If you can tell whether the question is counting trials, failures, or successes, you are already most of the way there.

## Connections

### Bernoulli Trial

Negative binomial distribution is built from repeated Bernoulli trials. Each trial has only two outcomes, success or failure, and the probability of success stays constant from trial to trial. If that setup is not true, the negative binomial model stops being a good fit.

### Geometric Distribution

Geometric distribution is the special case of the negative binomial distribution where you stop after the first success. Both deal with repeated independent trials with the same success probability. The difference is that negative binomial counts how long it takes to reach more than one success.

### Probability Mass Function (PMF)

The PMF tells you the probability of each exact outcome value. For negative binomial, the PMF gives the probability that the rth success happens at a certain trial number, or that a certain number of failures occurs first. The combination term in the formula counts the possible arrangements.

### [Memoryless Property](/college-intro-stats/key-terms/memoryless-property)

The negative binomial distribution does not have the memoryless property, even though the geometric distribution does. That means past failures still matter when you are counting up to the rth success. The process restarts only in the sense that each new trial has the same p, not because the whole wait time loses its history.

## On the AP Exam

A quiz or problem set will usually give you a word problem and ask whether negative binomial fits before you calculate anything. Your job is to identify the target number of successes, decide what X counts, and check that the trials are independent with a constant p. If the question asks for the probability of the 4th success happening on the 7th trial, you set up the negative binomial PMF rather than a binomial formula. If X is defined as failures before the rth success, make sure you do not mix up failures and total trials. Many points are lost from using the right model with the wrong counting rule. On free-response style homework, you may also need to explain why the distribution applies, not just compute the probability.

## Negative Binomial Distribution vs Geometric Distribution

Geometric distribution is about the number of trials until the first success. Negative binomial distribution extends that idea to the time or number of failures before a later success, like the third or fifth success. If the problem only cares about the first success, use geometric. If it waits for r successes, use negative binomial.

## Key Takeaways

- Negative binomial distribution counts repeated Bernoulli trials until you reach a fixed number of successes.
- The model only works when trials are independent and the success probability stays the same each time.
- Always check what the random variable is counting, because some versions count total trials and others count failures.
- Geometric distribution is the special case of negative binomial when the target number of successes is 1.
- The combination part of the PMF counts the different trial orders that can lead to the same stopping point.

## FAQs

### What is negative binomial distribution in Intro to Statistics?

It is a discrete probability distribution for a sequence of independent Bernoulli trials stopped when a fixed number of successes occurs. Depending on the textbook, X may count total trials needed or failures before the target success. The key idea is that the stopping point is based on reaching r successes, not on a fixed number of trials.

### How is negative binomial distribution different from geometric distribution?

Geometric distribution is the one-success version of the idea, so it finds the wait time until the first success. Negative binomial keeps going until the rth success, so it handles longer waiting periods. If you only need the first success, geometric is the better match.

### What does the negative binomial PMF tell you?

The PMF gives the probability of one exact outcome, such as the rth success happening on a specific trial or a specific number of failures occurring first. The formula combines the success and failure probabilities with a counting term that lists all the ways that sequence can happen. That is why combinations appear in the expression.

### When do I use negative binomial distribution on a homework problem?

Use it when the problem says to keep trying until a target number of successes happens. Phrases like “until the third sale,” “until two passes,” or “before the fifth success” are strong clues. If the problem fixes the number of trials instead, you are probably looking at binomial distribution instead.

## Related Study Guides

- [4.4 Geometric Distribution](/college-intro-stats/unit-4/4-geometric-distribution/study-guide/Ce9XX1u25hRwDMv3)

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