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

# Binomial characteristics vs negative binomial

Probability and Statistics · Chapter 2: Discrete Probability Distribution

- Asked in: 2074 Asoj, 4 marks
- Answer checked by Claude Sonnet 5.5 on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_5/probability_and_statistics/binomial-vs-nb-characteristics
- Whole chapter: https://www.bctengineer.com/md/semester_5/probability_and_statistics/chapter-2

## Question

What are the characteristics of Binomial Distribution and how does it differ from Negative Binomial Distribution?

## Short answer

Binomial: fixed n independent trials with constant p, counts successes; mean np, variance npq. Negative binomial: number of trials until k-th success, mean k/p, variance kq/p^2. The key difference: binomial fixes trials and counts successes; negative binomial fixes successes and counts trials.

## Answer

#### Characteristics of the binomial distribution
1. The experiment consists of a fixed number n of trials.
2. Each trial has only two outcomes, success (S) or failure (F).
3. The probability of success p is the same in every trial; q = 1 − p.
4. The trials are independent.
5. The random variable X counts the successes, so X = 0, 1, …, n.

P(X = x) = C(n, x) pˣ qⁿ⁻ˣ, mean = np, variance = npq (always less than the mean).

#### Negative binomial distribution
In repeated independent trials with constant p, let X be the trial on which the k-th success occurs:

P(X = x) = C(x − 1, k − 1) pᵏ qˣ⁻ᵏ, x = k, k + 1, …; mean = k/p, variance = kq/p².

#### Difference

| Basis | Binomial | Negative binomial |
|---|---|---|
| Number of trials | Fixed (n) | Random |
| Number of successes | Random | Fixed (k) |
| X measures | Successes in n trials | Trials until k-th success |
| Possible values | 0, 1, …, n | k, k+1, … (unbounded) |
| Mean | np | k/p |
| Variance | npq | kq/p² |

Example: the number of heads in 10 tosses is binomial; the number of tosses needed to get 3 heads is negative binomial.

## Check yourself

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

1. Which quantity is fixed in a binomial experiment but random in a negative binomial one?
   A. The number of successes
   B. The number of trials
   C. The probability of success p
   D. The number of failures only
   Answer: B. Binomial fixes n trials and counts successes; negative binomial fixes the number of successes k and the trial count is random. (Key point: Fixed vs random trials)

2. In a negative binomial distribution, X measures:
   A. The number of successes in n trials
   B. The number of failures before the first trial
   C. The proportion of successes in the sample
   D. The trial on which the k-th success occurs
   Answer: D. X is the number of trials needed to reach the k-th success. (Key point: What X measures)

3. For the binomial distribution, how does the variance compare with the mean?
   A. Variance npq is always less than the mean np
   B. Variance equals the mean
   C. Variance is greater than the mean
   D. Variance is k q/p^2 and mean is k/p
   Answer: A. Since q < 1, npq < np. (Key point: Mean and variance)

4. Which pair of ranges is correct?
   A. Binomial: 1, 2, ... unbounded; negative binomial: 0 to n
   B. Both take values 0 to n
   C. Binomial: 0 to n; negative binomial: k, k+1, ... unbounded
   D. Binomial: k, k+1, ...; negative binomial: 0 to n
   Answer: C. Binomial counts successes so is bounded by n; negative binomial counts trials, starting at k with no upper limit. (Key point: Range of values)

## Close variants set in other exams

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

## Related in Probability and Statistics

- [Negative binomial definition and characteristics](https://www.bctengineer.com/question.md?key=semester_5%2Fprobability_and_statistics%3Anb-definition-characteristics) — Probability and Statistics, chapter 2, asked 2×
- [Binomial vs negative binomial differences](https://www.bctengineer.com/question.md?key=semester_5%2Fprobability_and_statistics%3Anb-vs-binomial-diff-similarities) — Probability and Statistics, chapter 2, asked 4×
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- [Hypergeometric distribution; binomial approximation](https://www.bctengineer.com/question.md?key=semester_5%2Fprobability_and_statistics%3Ahypergeometric-define-binomial-approx) — Probability and Statistics, chapter 2, asked 2×
- [Gamma distribution: characteristics, applications](https://www.bctengineer.com/question.md?key=semester_5%2Fprobability_and_statistics%3Agamma-definition-characteristics) — Probability and Statistics, chapter 3

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