Conditional Probability

 $$P(A\;|\;B) = \text{Probability of $A$ given $B$}$$ 

Definition. If $P(B) \neq 0$, $P(A\;|\;B)= \frac{P(A \land B)}{P(B)}$

Product Rule. $P(A \land B) = P(B) P(A\;|\;B)$

General Product Rule.

 $$P(A_1 \land A_2 \land \cdots \land A_N) = P(A_1)P(A_2\;|\;A_1)P(A_3\;|\;A_1,A_2)\cdots P(A_N\;|\;A_1,A_2,\ldots,A_{N-1})$$ 

Independence

Definition. An event $A$ is independent of an event $B$ if $P(A\;|\;B)=P(A)$ or if $P(B)=0$

Theorem (Product Rule for Independent Events). If $A$ is independent of $B$, then $P(A \land B) = P(A) P(B)$

Theorem (Symmetry of Independence). If $A$ is independent of $B$, then $B$ is independent of $A$.