Conditional Probability

The conditional probability of $A$ given that $B$ has occurred, denoted $P(A\vert B)$, equals

$$P(A\vert B) = {P(A\cap B)\over P(B)},$$ (1)

which can be proven directly using a Venn Diagram. Multiplying through, this becomes

$$P(A\vert B)P(B) = P(A\cap B),$$ (2)

which can be generalized to

$$P(A\cup B\cup C) =P(A)P(B\vert A)P(C\vert A\cup B).$$ (3)

Rearranging (1) gives

$$P(B\vert A) = {P(B\cap A)\over P(A)}.$$ (4)

Solving (4) for $P(B\cap A) = P(A\cap B)$ and plugging in to (1) gives

$$P(A\vert B) = {P(A)P(B\vert A)\over P(B)}.$$ (5)

See also Bayes' Formula