Definition:Independent Events

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:

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

It follows from Event Independence is Symmetric that it follows directly that:

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

Thus it makes sense to refer to two events as independent of each other.

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:

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

Equivalence of Definitions

The fact that these definitions are equivalent follows directly from elementary algebra from the Product Rule for Probabilities:

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

General Definition

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

A set of events $\mathcal A = \left\{{A_i: i \in I}\right\}$ is independent iff, for all finite subsets $J$ of $I$:

$\displaystyle (1): \quad \Pr \left({\bigcap_{i \in J} A_i}\right) = \prod_{i \in J} \Pr \left({A_i}\right)$

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

Pairwise Independent

The above set $\mathcal A = \left\{{A_i: i \in I}\right\}$ is pairwise independent iff $(1)$ above holds whenever $\left|{J}\right| = 2$.

Note that although independence (in the general sense above) implies pairwise independence, a collection of events can be pairwise independent without being independent.

See Pairwise Independence does not imply Independence.

Dependent

If events are not independent, then they are dependent (on each other), and vice versa.

Sources

• Geoffrey Grimmett: Probability: An Introduction (1986): $\S 1.7$