Independence

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$A$ and $B$ are independent when

$$ P(B|A) = P(B) $$

Whether $A$ happens or not does not affect the probability of $B$.

By definition of $P(B|A)$,

$$ {P(A \cap B)} = P(A)P(B) $$

In this case, ${P(A \cap B)}$ is the joint and $P(A)P(B)$ is the marginal distributions.

If $A$ is independent of $B$, $B$ is also independent of $A$.

Furthermore, if $A$ and $B$ are independent, the following three are also independent.