Binomial Distribution

Maths: Statistics for machine learning

2 min read

Published Oct 22 2025, updated Oct 23 2025

Machine LearningMathsNumPyPandasPythonStatistics

The Binomial Distribution represents the probability of getting k successes in n independent Bernoulli trials,
where each trial has the same probability of success p.

In simple terms:

“If you repeat a yes/no experiment n times, what’s the probability of getting exactly k yes outcomes?”

Formula: Probability Mass Function (PMF)

Where:

X = number of successes
n = number of trials
k = specific number of successes (0 ≤ k ≤ n)
p = probability of success
number of ways to choose k successes:

The sum of all probabilities = 1:

Examples:

Coin toss - number of heads in 10 tosses
Email opens - number of users who open email
Loan approvals - number of approvals
Product purchase - number of buyers

A bar chart showing the probability of getting 0, 1, 2, …, n successes, with the highest bar near the mean (np).

Example (n=10, p=0.5):

Mean = 5 → distribution is centred around 5 successes
Symmetrical since p=0.5
Shape becomes skewed when p ≠ 0.5 (right- or left-skewed)

In Machine Learning

Modelling count of successes - e.g., number of customers who convert
Feature probability modelling - Discrete count features (e.g., binary outcomes repeated n times)
Bernoulli → Binomial extension - Multiple independent binary outcomes
Evaluation metrics - Used in hypothesis testing and confidence intervals for proportions