Two in­de­pen­dent events

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, $P(B\mid A)=P\left(B\right)$. Equivalently, $A$ and $B$ are independent if $P\left(A\right)$ doesn’t change if you condition on $B$: $P(A\mid B)=P\left(A\right)$.

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

All these definitions are equivalent:

$$P(A,B)=P\left(A\right)P(B\mid A)$$

by the chain rule, so

$$P(A,B)=P\left(A\right)P\left(B\right)\iff P\left(A\right)P(B\mid A)=P\left(A\right)P\left(B\right),$$

and similarly for $P\left(B\right)P(A\mid B)$.