The conditional probability of an event 
assuming that 
has occurred with 
, denoted 
, equals
 |
(1)
|
which can be illustrated using a Venn diagram. Multiplying through, this becomes
 |
(2)
|
which, for events 
, 
, and 
with 
, generalizes to
 |
(3)
|
When also 
,
interchange 
and 
in (2) to express 
in terms of the reverse conditional probability 
,
obtaining
 |
(4)
|
Solving (4) for the joint probability 
and substituting into
(3) gives Bayes' theorem,
 |
(5)
|
More generally, on a probability space 
,
conditional probability given a sigma-algebra 
is defined using conditional expectation
by
![P(A|G)=E[1_A|G],](/images/equations/ConditionalProbability/NumberedEquation6.svg) |
(6)
|
where 
is the indicator function of the event 
(Durrett 2019, p. 208). Conditioning on a random
variable 
means conditioning on the sigma-algebra generated
by 
.
When a regular version exists, these conditional probabilities can be chosen as a family 
such that, for each event 
, the function 
is 
-measurable and equals
almost surely. The function 
is then a probability measure on 
for almost every 
.
Such a family is called a regular conditional probability (Durrett 2019, p. 214).
