Basic Notions of Probability Theory - Probabilistic Composition of Preferences, Theory and Applications - page 118

So, the concept of probability involves three elements: the sample space S, the
set of events

Λ (S) and the probability function p. A triple (S,

ʛ , p), where ʛ is a

sigma algebra of parts of S and p is a probability with domain ʛ is called a

probability space.
Some properties follow immediately from the de nition of probability. For

instance,

p ð Ø Þ ¼ 0 ; for any event A

;
denoting by A C the complement of A with respect to S,

pA ð ;

pA ðÞ ¼ 1

for any two events A and B,

p ð A \ B Þ ¼ pA ðÞ pA ð n B Þ;

A B ! pA ðÞ pB ðÞ

and

p AUB ð Þ ¼ pA ð n B Þþ pA ð \ B Þþ pB ð n A Þ:

ʛ ʛ

(S) with p(B) > 0, it is
possible to derive another probability on S that will coincide with p if and only if
p(B) = 1. This new probability is called the probability p conditional on B. This
probability is denoted p(|B) and the probability of any event A of ʛ (S) by p(|B) is
given by

For every probability p on (S, (S)) and any event B of

pA ð j B Þ ¼ pA ð \ B Þ= pB ðÞ:

An important use of conditional probabilities involves conditioning separately
on the elements of a partition. It employs the following property.

Total Probability Theorem
For any countable partition Bi i } i ∈ N of S with Bi i ∈ ʛ (S) and p(Bi) > 0 for all i, and

for any A ∈ ʛ (S),

X

pA ðÞ ¼ pA ð j B i Þ pB ð :

i 2 N

Here the reader must remember that a partition of a set S is any collection of
mutually excludent subsets of S whose union is S; sets are mutually excludent if
and only if they are pairwise disjoint, what means that the intersection of any pair of
them is empty.

Independence
With respect to a probability p for which p(A) ≠ 0 ≠ p(B),

two events A and B are independent $ pAB ð j Þ ¼ PA ðÞ: