Equivalence of Definitions of Independent Events

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

$\blacksquare$


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