Introduction to Sample Space and Probability - Random Signals and Processes Primer with MATLAB - page 9

If the event A is defined as in ( 1.5 ), then event A has three elements resulting in:

PfAg¼ 3 = 6 ¼ 1 = 2 : (1.36)

1.4 Relative Frequency Definition of Probability

A random experiment must be, in principle, repeatable an arbitrary number of times
under the same conditions [HEL91, p. 17]. Consider that the experiment has been
repeated N times and the number of times that event A occurred in N trials is
denoted as n A . Define the relative frequency q ( A ) as:

n A
N :

qðAÞ¼ (1.37)

This property, called statistical regularity , implies that if the number of trials is
increased the relative frequency varies less and less and approaches the probability
of event A ,

n A
N ! PfAg;

qðAÞ¼ as N !1: (1.38)

It should be noted that observations of the statistical stability of relative fre-
quency (statistical regularity) of many phenomena served as the starting point for
developing a mathematical theory of probability [GNE82, p. 45].
The probability defined in ( 1.38 ), obeys Axioms I-III described in Sect. 1.2.1 ,as
shown in the following.

Axiom I From ( 1.38 ), it is obvious that the numbers n A and N are always positive
and thus the probability P { A } cannot be negative.

Axiom II If in all trials the event A occurs, then n A ¼ N , and A is a certain event,
and

PfAg¼N=N ¼ 1 :

(1.39)

Axiom III If two events A and B are mutually exclusive ( A\B ) ¼ 0, then

n AþB ¼ n A þ n B ; (1.40)

and the probability is equal to:

PfA [ Bg¼PfA þ Bg¼n A þB =N ¼ n A =N þ n B =N ¼ PfAgþPfBg: (1.41)