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

# Binomial definition and conditions

Probability and Statistics · Chapter 2: Discrete Probability Distribution

- Asked in: 2075 Asoj, 2+2 marks
- Video: [An Introduction to the Binomial Distribution](https://www.youtube.com/watch?v=qIzC1-9PwQo) — jbstatistics
- Answer checked by Claude Sonnet 5.5 on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_5/probability_and_statistics/binomial-definition-conditions
- Whole chapter: https://www.bctengineer.com/md/semester_5/probability_and_statistics/chapter-2

## Question

Define binomial distribution and explain the condition for Binomial distribution.

## Short answer

A binomial distribution gives the probability of x successes in n independent Bernoulli trials with constant success probability p, as P(X=x)=C(n,x)p^x q^(n-x). Conditions: fixed number of trials, two outcomes per trial, independence, constant p.

## Answer

#### Definition
A random variable X that counts the number of successes in n independent Bernoulli trials, each with the same success probability p, has a **binomial distribution** with parameters n and p, written X ~ B(n, p):

P(X = x) = C(n, x) pˣ qⁿ⁻ˣ, x = 0, 1, 2, …, n, q = 1 − p.

Mean = np and variance = npq.

#### Conditions for the binomial distribution
1. The number of trials n is fixed in advance.
2. Each trial has exactly two outcomes, success or failure.
3. The trials are independent of each other.
4. The probability of success p is constant in every trial.

Example: the number of defective bulbs in a random sample of 10 taken from a large production line, where each bulb is defective with probability 0.05.

## Check yourself

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

1. X ~ B(n, p) is the distribution of:
   A. The number of trials until the first success
   B. The number of rare events in an interval
   C. The number of successes in n independent trials with constant p
   D. The number of successes in draws without replacement from a finite population
   Answer: C. A binomial counts successes in n independent Bernoulli trials with the same p. (Key point: Definition)

2. Which is NOT a condition for a binomial distribution?
   A. The number of trials is random
   B. The trials are independent
   C. Each trial has two outcomes
   D. The success probability is constant
   Answer: A. The number of trials n must be fixed in advance. (Key point: Conditions)

3. The mean and variance of B(n, p) are:
   A. np and np
   B. n/p and nq/p^2
   C. pq and npq
   D. np and npq
   Answer: D. Mean = np, variance = npq with q = 1 - p. (Key point: Mean and variance)

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