Conditional Probability and Independent Events

Sometimes new information changes the probability of an event — knowing that a customer is a repeat buyer changes the probability they will buy again this month. The conditional probability of event $A$, given that event $B$ has already occurred, is written $P(A\mid B)$ and defined as:

Rearranging this definition gives the multiplication theorem of probability, used to find the probability that both $A$ and $B$ occur:

Independent events are a special case: two events $A$ and $B$ are independent if the occurrence of one has genuinely no effect on the probability of the other — that is, $P(A\mid B) = P(A)$ and $P(B\mid A)=P(B)$. Substituting this into the multiplication theorem gives the simplified rule used whenever two events are known (or assumed) to be independent: