The conditional probability of an event 
assuming that 
has occurred, denoted 
, equals
 |
(1)
|
which can be proven directly using a Venn diagram. Multiplying through, this becomes
 |
(2)
|
which can be generalized to
 |
(3)
|
To express 
in terms of the reverse conditional probability 
, interchange 
and 
in (2) to obtain
 |
(4)
|
Solving (4) for the joint probability 
and substituting into
(3) gives Bayes' theorem,
 |
(5)
|
