Introduction to Sample Space and Probability - Random Signals and Processes Primer with MATLAB - page 5

The joint event or intersection of events A and B -denoted as A\B -is an event
which contains points common to A and B (i.e., the event “both A and B occurred”).
The intersection is sometimes called the product of events and is denoted as AB . For
example, the joint event of A and B from Fig. 1.1a is a null event, A\B ¼ {0},
because the events A and B cannot occur simultaneously.
If events A and B do not have elements in common, A\B ¼ 0, then they are said
to be disjoint ,or mutually exclusive . The events A and B from Fig. 1.1a are mutually
exclusive.
Figure 1.4 is a Venn diagram representation of different operations with events.

1.2 Probability of Events

1.2.1 Axioms

The probability of event A is a real number assigned to event A -denoted as
P { A }-which satisfies the following axioms, formulated by the Russian mathemati-
cian A.N. Kolmogorov:

Axiom I PfAg 0 : (1.11)

The axiom states that the probability of event A is a nonnegative number.

Axiom II

PfSg¼ 1 : (1.12)

The axiom states that the sample space is itself an event which always occurs.

Axiom III The axiom states if two events A and B are mutually exclusive (i.e.,
A\B ¼ 0) then the probability of either event A ,or B occurring, P { A[B }, is the
sum of their probabilities:

PfA [ Bg¼PfA þ Bg¼PfAgþPfBg:

(1.13)

Since the word axiom means self-evident statement [MIL04, p. 12], the proof for
( 1.11 )-( 1.13 ) is omitted.

Example 1.2.1 In a coin tossing experiment, the events A ¼ {H} and B ¼ {T} are
mutually exclusive (i.e., only “heads” or “tails” can occur), resulting in:

Pf H [ T g¼Pf H gþPf T g¼ 1 : (1.14)

The corresponding sample space is:

S ¼ A [ B (1.15)

and from ( 1.14 ) we have:

PfSg¼ 1 : (1.16)