Binomial Distribution

The binomial distribution models the probability of a fixed number of successes in repeated independent trials.

What is binomial distribution?

The binomial distribution is a discrete probability distribution that describes the number of successes in a fixed number of independent trials, where each trial has only two possible outcomes: success or failure.

Conditions for binomial distribution

A random experiment follows a binomial distribution if:

The number of trials (n) is fixed
Each trial has two outcomes (success or failure)
The probability of success (p) is constant
Trials are independent

Binomial distribution formula

The probability of observing exactly k successes in n trials is:

P(X = k) = C(n, k) × pᵏ × (1 − p)ⁿ⁻ᵏ

Example

Tossing a fair coin 4 times:

n = 4
k = 2
p = 0.5

P(X = 2) = 6 × (0.5)² × (0.5)² = 0.375

How to interpret the result

The result represents the probability of obtaining exactly k successes out of n trials. Higher probabilities indicate more likely outcomes.

Why is binomial distribution important?

Binomial distribution is widely used in statistics, quality control, data science, finance, and experimental analysis. It forms the foundation for more advanced probability models.