# Equivalence of Definitions of Independent Events

## Theorem

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 $\Pr \left({B}\right) > 0$.

The following definitions of the concept of **Independent Events** are equivalent:

### 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:

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

## Proof

\(\displaystyle \Pr \left({A \mid B}\right)\) | \(=\) | \(\displaystyle \Pr \left({A}\right)\) | |||||||||||

\(\displaystyle \iff \ \ \) | \(\displaystyle \dfrac {\Pr \left({A \cap B}\right)} {\Pr \left({B}\right)}\) | \(=\) | \(\displaystyle \Pr \left({A}\right)\) | by definition of Conditional Probability | |||||||||

\(\displaystyle \iff \ \ \) | \(\displaystyle \Pr \left({A \cap B}\right)\) | \(=\) | \(\displaystyle \Pr \left({A}\right) \Pr \left({B}\right)\) | valid as $\Pr \left({B}\right) > 0$ |

$\blacksquare$

## Sources

- 1986: Geoffrey Grimmett and Dominic Welsh:
*Probability: An Introduction*... (previous) ... (next): $\S 1.7$: Independent Events