Two independent events

$$ \newcommand{\bP}{\mathbb{P}} $$

We say that two events, $A$ and $B$, are independent when learning that $A$ has occurred does not change your probability that $B$ occurs. That is, $\bP(B \mid A) = \bP(B)$. Equivalently, $A$ and $B$ are independent if $\bP(A)$ doesn’t change if you condition on $B$: $\bP(A \mid B) = \bP(A)$.

Another way to state independence is that $\bP(A,B) = \bP(A) \bP(B)$.

All these definitions are equivalent:

$$\bP(A,B) = \bP(A)\; \bP(B \mid A)$$

by the chain rule, so

$$\bP(A,B) = \bP(A)\; \bP(B)\;\; \Leftrightarrow \;\; \bP(A)\; \bP(B \mid A) = \bP(A)\; \bP(B) \ ,$$

and similarly for $\bP(B)\; \bP(A \mid B)$.