Independence

$A$ 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.