B. P. Lathi, Zhi Ding - Modern Digital and Analog Communication Systems-Oxford University Press (2009)

8.1 Concept of Probability 395
Figure 8.2
Representation of
(a) complement,
(b) union, and
(c) intersection of
events.
s
s
s
(a)
A'
AUB = BUA
(b)
AB and A c B
(c)
Observe that the union operation commutes:
AUB=BUA (8.1)
The intersection of events A and B, denoted by A nB or simply by AB, is the event that contains
points common to A and B. This is the event that "outcome is both A and B," also known as
the joint event A n B. Thus, the event AeB, "a number that is even and equal to or less than 4
is thrown," is a set (s2, s4), and similarly for A 0 B,
Observe that the intersection also commutes
AnB=BnA (8.2)
All these concepts can be demonstrated on a Venn diagram (Fig. 8.2). If the events A and
B are such that
AnB=0 (8.3)
then A and B are said to be disjoint, or mutually exclusive, events. This means events A and
B cannot occur simultaneously. In Fig. 8.1 events A e and A 0 are mutually exclusive, meaning
that in any trial of the experiment if A e occurs, A 0 cannot occur at the same time, and vice
versa.
Relative Frequency and Probability
Although the outcome of an experiment is unpredictable, there is a statistical regularity about
the outcomes. For example, if a coin is tossed a large number of times, about half the times
the outcome will be "heads," and the remaining half of the times it will be "tails." We may
say that the relative frequency of the two outcomes "heads" or "tails" is one-half. This relative
frequency represents the likelihood of a particular event.
Let A be one of the events of interest in an experiment. If we conduct a sequence of N
independent trials* of this experiment, and if the event A occurs in N (A) out of these N trials,
then the fraction
f (A) = lim N(A)
N--HX! N
(8.4)
* Trials conducted under similar discernible conditions.