Independence of two events

Remarks

If $P(B) > 0$, then $A$ and $B$ are independent if and only if $P(A \mid B) = P(A)$, with the conditional probability. If $A$ and $B$ are independent, then so are $A$ and $B^c$, $A^c$ and $B$, and $A^c$ and $B^c$. An event of probability $0$ or $1$ is independent of each event. An event $A$ is independent of itself if and only if $P(A) \in \{0, 1\}$. Independence is not the same as disjointness. Disjoint events $A$ and $B$ with $P(A) > 0$ and $P(B) > 0$ are never independent.