---
title: "Binomial vs negative binomial differences"
subject: "Probability and Statistics"
chapter: 2
key: "semester_5/probability_and_statistics:nb-vs-binomial-diff-similarities"
source: https://www.bctengineer.com/q/semester_5/probability_and_statistics/nb-vs-binomial-diff-similarities
site: BCT Engineering
---

# Binomial vs negative binomial differences

Probability and Statistics · Chapter 2: Discrete Probability Distribution

- Asked in: 2072 Chaitra, 2+3 marks; 2071 Chaitra, 2+3 marks; 2070 Chaitra, 2+3 marks; 2069 Chaitra, 2+2 marks (asked 4 times)
- Video: [Binomial vs Negative Binomial vs Geometric Distributions](https://www.youtube.com/watch?v=VlVDKA9pg4A) — Explained by Michael
- Answer checked by Claude Sonnet 5.5 on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_5/probability_and_statistics/nb-vs-binomial-diff-similarities
- Whole chapter: https://www.bctengineer.com/md/semester_5/probability_and_statistics/chapter-2

## Question

Write the differences and similarities between Binomial and Negative Binomial Distribution.

## Short answer

Binomial has fixed trials and counts successes; negative binomial fixes successes and counts trials. Both use independent Bernoulli trials with constant p and are discrete.

## Answer

#### Definitions
- **Binomial distribution:** the number of successes X in a **fixed number n** of independent trials, each with success probability p.
  P(X = x) = C(n, x) pˣ qⁿ⁻ˣ, x = 0, 1, …, n; mean = np, variance = npq.
- **Negative binomial distribution:** the number of trials X needed to get a **fixed number r of successes**.
  P(X = x) = C(x − 1, r − 1) pʳ qˣ⁻ʳ, x = r, r + 1, r + 2, …; mean = r/p, variance = rq/p².

#### Differences

| Basis | Binomial | Negative binomial |
|---|---|---|
| Fixed quantity | Number of trials n | Number of successes r |
| Random variable | Number of successes in n trials | Number of trials to get r successes |
| Range of X | 0 to n (finite) | r, r + 1, … (infinite) |
| pmf | C(n, x) pˣ qⁿ⁻ˣ | C(x − 1, r − 1) pʳ qˣ⁻ʳ |
| Mean | np | r/p |
| Variance | npq (always less than the mean) | rq/p² (greater than the mean when q > p) |
| Last trial | No condition | The last trial must be a success |
| Example | Heads in 10 tosses | Tosses needed to get 3 heads |

#### Similarities
- Both are based on independent Bernoulli trials with only two outcomes (success or failure).
- Both have a constant probability of success p in every trial, with q = 1 − p.
- Both are discrete distributions.
- When r = 1, the negative binomial becomes the geometric distribution, the waiting time for the first success in a Bernoulli process, which is the same process that underlies the binomial.
- Both can be approximated by the Poisson or normal distribution under suitable conditions.

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. What is fixed in the negative binomial but not the binomial?
   A. Number of trials
   B. Probability p
   C. Number of outcomes
   D. Number of successes r
   Answer: D. Binomial fixes n; negative binomial fixes r. (Key point: Fixed quantity)

2. What is the range of X in negative binomial?
   A. 0 to n
   B. r, r+1, r+2, ...
   C. 1 to r
   D. 0 or 1
   Answer: B. The trials needed are at least r and unbounded. (Key point: Range)

3. Which statement is correct?
   A. Both variances equal the mean
   B. Binomial variance < mean, NB variance > mean (q>p)
   C. NB variance < mean
   D. Binomial variance > mean
   Answer: B. npq < np, while rq/p^2 > r/p when q > p. (Key point: Variance vs mean)

## Related in Probability and Statistics

- [Binomial characteristics vs negative binomial](https://www.bctengineer.com/question.md?key=semester_5%2Fprobability_and_statistics%3Abinomial-vs-nb-characteristics) — Probability and Statistics, chapter 2, asked 4×
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