Event Independence is Symmetric

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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.


We assume throughout that $\map \Pr A > 0$ and $\map \Pr B > 0$.

Let $A$ be independent of $B$.

Then by definition:

$\condprob A B = \map \Pr A$

From the definition of conditional probabilities, we have:

$\condprob A B = \dfrac {\map \Pr {A \cap B} } {\map \Pr B}$

and also:

$\condprob B A = \dfrac {\map \Pr {A \cap B} } {\map \Pr A}$

So if $\condprob A B = \map \Pr A$ we have:

\(\ds \map \Pr A\) \(=\) \(\ds \frac {\map \Pr {A \cap B} } {\map \Pr B}\)
\(\ds \leadsto \ \ \) \(\ds \map \Pr B\) \(=\) \(\ds \frac {\map \Pr {A \cap B} } {\map \Pr A}\)
\(\ds \leadsto \ \ \) \(\ds \map \Pr B\) \(=\) \(\ds \condprob B A\)

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