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:
 * $$\Pr \left({A | B}\right) = \Pr \left({A}\right)$$

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

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

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

$$ $$ $$

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