# Definition:Independent Events

## Contents

## Definition

Let $\mathcal E$ be an experiment with probability space $\left({\Omega, \Sigma, \Pr}\right)$.

Let $A, B \in \Sigma$ be events of $\mathcal E$ such that $\Pr \left({A}\right) > 0$ and $\Pr \left({B}\right) > 0$.

### Definition 1

The events $A$ and $B$ are defined as **independent (of each other)** iff the occurrence of one of them does not affect the probability of the occurrence of the other one.

Formally, $A$ is independent of $B$ iff:

- $\Pr \left({A \mid B}\right) = \Pr \left({A}\right)$

where $\Pr \left({A \mid B}\right)$ denotes the conditional probability of $A$ given $B$.

### Definition 2

The events $A$ and $B$ are defined as **independent (of each other)** iff the occurrence of both of them together has the same probability as the product of the probabilities of each of them occurring on their own.

Formally, $A$ and $B$ are independent iff:

- $\Pr \left({A \cap B}\right) = \Pr \left({A}\right) \Pr \left({B}\right)$

## General Definition

The definition can be made to apply to more than just two events.

Let $\mathcal A = \left\{{A_i: i \in I}\right\}$ be a family of events of $\mathcal E$.

Then $\mathcal A$ is **independent** iff, for all finite subsets $J$ of $I$:

- $\displaystyle \Pr \left({\bigcap_{i \mathop \in J} A_i}\right) = \prod_{i \mathop \in J} \Pr \left({A_i}\right)$

That is, if the occurrence of any finite collection of $\mathcal A$ has the same probability as the product of each of those sets occurring individually.

### Pairwise Independent

Let $\mathcal A = \left\{{A_i: i \in I}\right\}$ be a set of events of $\mathcal E$.

Then $\mathcal A$ is **pairwise independent** iff:

- $\forall j, k \in I: \Pr \left({A_j \cap A_k}\right) = \Pr \left({A_j}\right) \Pr \left({A_k}\right)$

That is, if every pair of events of $\mathcal E$ are independent of each other.

That is, $\mathcal A$ is **pairwise independent** iff the condition for general independence:

- $\displaystyle \Pr \left({\bigcap_{i \mathop \in J} A_i}\right) = \prod_{i \mathop \in J} \Pr \left({A_i}\right)$

holds whenever $\left|{J}\right| = 2$.

## Dependent

If $A$ and $B$ are not independent, then they are **dependent (on each other)**, and vice versa.

## Also see

- Event Independence is Symmetric: thus it makes sense to refer to two events as
**independent of each other**.