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

396 FUNDAMENTALS OF PROBABILITY THEORY
is called the relative frequency of the event A. Observe that for small N, the fraction N (A)/ N
may vary widely with N. As N increases, the fraction will approach a limit because of statistical
regularity.
The probability of an event has the same connotations as the relative frequency of that
event. Hence, we estimate the probability of each event, as the relative frequency of that event.*
Therefore, to an event A, we assign the probability P(A) as
P(A) = Jim N (A)
N---+ oo N
(8.5)
From Eq. (8.5), it follows that
0 :S P(A) :S I
(8.6)
Example 8.1 Assign probabilities to each of the six outcomes in Fig. 8.1.
Because each of the six outcomes is equally likely in a large number of independent trials,
each outcome will appear in one-sixth of the trials. Hence,
1
P((;) = - 6
i = 1, 2, 3, 4, 5, 6 (8.7)
Consider now the two events A and B of an experiment. Suppose we conduct N independent
trials of this experiment and events A and B occur in N (A) and N (B) trials, respectively. If A
and B are mutually exclusive (or disjoint), then if A occurs, B cannot occur, and vice versa.
Hence, the event A U B occurs in N (A) + N (B) trials and
N(A) +N(B)
P(A U B) = lim
N---+oo N
= P(A) + P(B) if A n B = 0 (8.8)
This result can be extended to more than two mutually exclusive events. In other words, if
events {A;} are mutually exclusive such that
then
A; n Aj = 0 i I- j
* Observe that we are not defining the probability by the relative frequency. To a given event, a probability is closely
estimated by the relative frequency of the event when this experiment is repeated many times. Modern theory of
probability. being a branch of mathematics, starts with certain axioms about probability [Eqs. (8.6), (8.8), and (8.11)].
It assumes that somehow these probabilities are assigned by nature. We use relative frequency to estimate probability
because it is reasonable in the sense that it closely approximates our experience and expectation of "probability."