et
and
be two events of the same event space
,
where
is the total event space,
is a collection of subsets of
equipped with set operations and
is a set function acting
with the
properties
for any at most countable collection of sets
such
that
.
Conditional probability
of
conditionally on
is defined as


(Bayes formula)

If the whole space
is represented as some disjoint union
then for any


(Total probability rule)

The last relationship is called "the total probability rule".
Conditional probability has transitive property:
hence,


(Transitive bayes formula)

Suppose we are facing calculation of the quantity
and the quantity
is easily computable. The repeated application of the Bayes formula expresses
in terms of
.
Indeed, the Bayes formula is symmetrical:
hence
The properties
(
Total_probability_rule
),(
Transitive_Bayes_formula
)
and (
Inversion_remark
) are used repeatedly in
the following chapters.
