# 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)** if and only if the occurrence of one of them does not affect the probability of the occurrence of the other one.

Formally, $A$ is independent of $B$ if and only if:

- $\map \Pr {A \mid B} = \map \Pr A$

where $\map \Pr {A \mid B}$ denotes the conditional probability of $A$ given $B$.

### Definition 2

The events $A$ and $B$ are defined as **independent (of each other)** if and only if 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:

- $\map \Pr {A \cap B} = \map \Pr A \, \map \Pr B$

## General Definition

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

Let $\AA = \family {A_i}_{i \mathop \in I}$ be an indexed family of events of $\EE$.

Then $\AA$ is **independent** if and only if, for all finite subsets $J$ of $I$:

- $\displaystyle \map \Pr {\bigcap_{i \mathop \in J} A_i} = \prod_{i \mathop \in J} \map \Pr {A_i}$

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

### Pairwise Independent

Let $\mathcal A = \family {A_i}_{i \mathop \in I}$ be an indexed family of events of $\EE$.

Then $\AA$ is **pairwise independent** if and only if:

- $\forall j, k \in I: \map \Pr {A_j \cap A_k} = \map \Pr {A_j} \, \map \Pr {A_k}$

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

That is, $\AA$ is **pairwise independent** if and only if the condition for general independence:

- $\displaystyle \map \Pr {\bigcap_{i \mathop \in J} A_i} = \prod_{i \mathop \in J} \map \Pr {A_i}$

holds whenever $\card J = 2$.

## 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**.