Independence

In Probability,

$A$ 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.

1. $A & B^C$
2. $A^C & B$
3. $A^C & B^C$