Multiplication Theorem

Rearranging the conditional probability formula $P(B/A)=P(A\cap B)/P(A)$ for $P(A\cap B)$ gives the multiplication theorem: for any two events $A$ and $B$ on a sample space $S$, $P(A\cap B)=P(A)\cdot P(B/A),$ and symmetrically, from $P(A/B)=P(A\cap B)/P(B)$, also $P(A\cap B)=P(B)\cdot P(A/B)$. This is the natural way to compute the probability that two events happen one after the other - especially useful when the second draw depends on what happened in the first, as in drawing without replacement. …