Event Independence is Symmetric

Theorem
Let $A$ and $B$ be events in a probability space.

Let $A$ be independent of $B$.

Then $B$ is independent of $A$.

That is, is independent of is a symmetric relation.

Proof
We assume throughout that $\Pr \left({A}\right) > 0$ and $\Pr \left({B}\right) > 0$.

Let $A$ be independent of $B$.

Then by definition:
 * $\Pr \left({A \mid B}\right) = \Pr \left({A}\right)$

From the definition of conditional probabilities, we have:
 * $\Pr \left({A \mid B}\right) = \dfrac{\Pr \left({A \cap B}\right)} {\Pr \left({B}\right)}$

and also:
 * $\Pr \left({B \mid A}\right) = \dfrac{\Pr \left({A \cap B}\right)} {\Pr \left({A}\right)}$

So if $\Pr \left({A \mid B}\right) = \Pr \left({A}\right)$ we have:

So by definition, $B$ is independent of $A$.