# Equivalence of Definitions of Independent Events

## Theorem

Let $\EE$ be an experiment with probability space $\struct {\Omega, \Sigma, \Pr}$.

Let $A, B \in \Sigma$ be events of $\EE$ such that $\map \Pr A > 0$ and $\map \Pr B > 0$.

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)** 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:

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

## Proof

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

\(\displaystyle \leadstoandfrom \ \ \) | \(\displaystyle \dfrac {\map \Pr {A \cap B} } {\map \Pr B}\) | \(=\) | \(\displaystyle \map \Pr A\) | Definition of Conditional Probability | |||||||||

\(\displaystyle \leadstoandfrom \ \ \) | \(\displaystyle \map \Pr {A \cap B}\) | \(=\) | \(\displaystyle \map \Pr A \, \map \Pr B\) | which is valid, as $\map \Pr B > 0$ |

$\blacksquare$

## Sources

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