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 $\map \Pr A > 0$ and $\map \Pr B > 0$.

Let $A$ be independent of $B$.

Then by definition:
 * $\map \Pr {A \mid B} = \map \Pr A$

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

and also:
 * $\map \Pr {B \mid A} = \dfrac {\map \Pr {A \cap B} } {\map \Pr A}$

So if $\map \Pr {A \mid B} = \map \Pr A$ we have:

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