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

# Binomial characteristics vs hypergeometric

Probability and Statistics · Chapter 2: Discrete Probability Distribution

- Asked in: 2076 Chaitra, 5 marks
- Video: [Overview of Some Discrete Probability Distributions (Binomial,Geometric,Hypergeometric,Poisson,NegB)](https://www.youtube.com/watch?v=UrOXRvG9oYE) — jbstatistics
- Answer checked by Claude Sonnet 5.5 on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_5/probability_and_statistics/binomial-vs-hypergeometric
- 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 Hypergeometric Distribution?

## Short answer

Binomial: fixed trials, constant p, independent. Hypergeometric: sampling without replacement, dependent trials. Variance differs by finite population correction.

## Answer

The **binomial distribution** gives the probability of x successes in n independent trials, each with the same success probability p:

P(X = x) = C(n, x) pˣ qⁿ⁻ˣ, x = 0, 1, …, n, where q = 1 − p.

#### Characteristics of the binomial distribution
1. The experiment has a fixed number of trials n.
2. Each trial has only two outcomes: success or failure.
3. The probability of success p is the same in every trial.
4. The trials are independent (sampling with replacement, or from an infinite population).
5. Mean = np, variance = npq, so variance < mean.
6. It is symmetric when p = 0.5, skewed right when p < 0.5 and skewed left when p > 0.5.

#### Hypergeometric distribution
When n items are drawn **without replacement** from a finite population of N items containing K successes:

P(X = x) = C(K, x) · C(N − K, n − x) / C(N, n)

Mean = nK/N, variance = n (K/N)(1 − K/N)(N − n)/(N − 1).

#### Difference

| Basis | Binomial | Hypergeometric |
|---|---|---|
| Sampling | With replacement (or infinite population) | Without replacement from a finite population |
| Trials | Independent | Dependent |
| p of success | Constant in every trial | Changes after every draw |
| Parameters | n, p | N, K, n |
| Mean | np | n(K/N) |
| Variance | npq | npq (N − n)/(N − 1), with p = K/N |
| Use | Coin tosses, defects in large production | Cards, small lots, committee selection |

When N is large compared with n (n/N < 0.05), the hypergeometric distribution is well approximated by the binomial with p = K/N.

## Check yourself

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

1. Which is NOT a binomial characteristic?
   A. Constant p
   B. Fixed number of trials
   C. p changes after every draw
   D. Independent trials
   Answer: C. p changing after each draw is a hypergeometric feature. (Key point: Binomial conditions)

2. How does hypergeometric sampling differ?
   A. Fixed r successes
   B. Infinite population
   C. Without replacement from a finite population
   D. With replacement
   Answer: C. Trials are dependent. (Key point: Sampling)

3. When is hypergeometric approximated by binomial?
   A. n larger than N
   B. p=0.5 only
   C. N large compared with n (n/N < 0.05)
   D. Never
   Answer: C. Removal of items barely changes p, so p = K/N works. (Key point: Approximation)

## 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 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×

## Related in Probability and Statistics

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