Definition:Independent Events

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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