## Bayes' Formula

Let and be Sets. Conditional Probability requires that

 (1)

where denotes Intersection (and''), and also that
 (2)

and
 (3)

Since (2) and (3) must be equal,
 (4)

From (2) and (3),
 (5)

Equating (5) with (2) gives
 (6)

so
 (7)

Now, let
 (8)

so is an event in and for , then
 (9)

 (10)

From (5), this becomes
 (11)

so
 (12)