Introduction to Sample Space and Probability - Random Signals and Processes Primer with MATLAB - page 6

1.2.2 Properties

From the Axioms, I, II, and III, we can derive more properties of probability as
shown here:

P. 1 Event A and its complement A are mutually exclusive. Consequently, the union
of A and A is a certain event with probability 1.

Proof From Axiom III, we have:

1 ¼ PfA [ Ag¼PfAgþPfAg: (1.17)

From here, the probability of the complement of event A is:

PfAg¼ 1 PfAg: (1.18)

P.2 Probability of event A is:

0 PfAg 1 : (1.19)

Proof From Axiom I the minimum probability of event A is equal to 0, this
corresponds to the maximum value of the probability of its complement, which is
equal to 1 (see ( 1.18 )). Similarly, from Axiom I, the minimum value of the
probability of the complement of A is equal to 0. In this case, from ( 1.18 ) it follows
that the maximum value of P { A } is equal to 1.
P.3 If B is a subset of event A (Fig. 1.5 ), B A , then,

PfBgPfAg:

(1.20)

Proof From Fig. 1.5 , we found:

A ¼ B [ B;
PfAg¼PfBgþPfBg;
PfBg¼PfAgPfBg:

(1.21)

Given that PfBg 0, thus (1.20) follows.

P.4 Probability of null events (empty events) is zero

Pf 0 g¼ 0 : (1.22)

Proof This property follows the property P.1 and Axiom II, recognizing that the
null event is the complement of the certain event S and thus:

Pf 0 g¼ 1 PfSg¼ 1 1 ¼ 0 : (1.23)