Conditional Probability and the Multiplication Theorem

Conditional probability asks: given that event $B$ has already occurred, what is the probability that $A$ also occurs? It is written $P(A|B)$ ('probability of $A$ given $B$') and defined as:

Symmetrically,

Rearranging either definition gives the Multiplication Theorem of Probability, which is how $P(A\cap B)$ is actually computed whenever the relevant conditional probability is easier to state than the joint probability directly:

$P(A\cap B)=P(A)\times P(B|A)=P(B)\times P(A|B)$ …