Introduction

The binomial distribution describes the number of successes in a fixed number of independent trials with two possible outcomes.

Conditions & Prerequisites

This section outlines the essential conditions that must be met for a random variable to follow a binomial distribution.

Conditions

The section outlines the essential conditions that define when a random variable follows a binomial distribution.

Notation

This section introduces the notation used in binomial distributions, explaining the relevant variables and symbols.

Binomial Probability Formula

The Binomial Probability Formula calculates the probability of achieving exactly k successes in n independent Bernoulli trials with a constant probability of success.

Derivation Idea

This section explains the derivation of the binomial probability formula by considering the number of ways to choose successes and the probability of a specific outcome.

Key Properties

This section outlines the key properties of the binomial distribution, including mean, variance, and standard deviation.

Cumulative Probabilities

Cumulative probabilities assess the likelihood of achieving a specific number of successes in binomial experiments.

Worked Examples

This section presents worked examples for understanding the binomial distribution, demonstrating specific scenarios and calculations.

Example 1 – Single Probability

Example 2 – Cumulative Probability

This section focuses on calculating cumulative probabilities in a binomial distribution, particularly 'at most’ and ‘at least’ scenarios.

Example 3 – Mean & Variance

This section covers the calculation of mean, variance, and standard deviation for a binomial distribution.

Example 4 – At Least 𝑘

This section explains how to calculate the probability of obtaining at least a certain number of successes in a binomial distribution.

Approximations For Large 𝑛

This section explains how to approximate the binomial distribution with a normal distribution when the number of trials, 𝑛, is large and the probability of success, 𝑝, is not close to 0 or 1.

Ib‐style Problem

This section presents an IB-style problem involving the binomial distribution, specifically related to a multiple-choice quiz scenario.

When Not To Use Binomial

This section outlines scenarios where the binomial distribution is not applicable, including non-independent trials, varying probabilities, sampling methods, and multiple outcomes.

Tips For Ib Exams

This section provides essential strategies for effectively approaching IB exams involving binomial distributions.

Summary

The binomial distribution quantifies the number of successes in a given number of independent trials with fixed probabilities for success and failure.