Page 21 - Probability, Random Variables and Random Processes

CHAP. 11 PROBABILITY

Determine the events

U B, and OB,
i= 1 i= 1
(a) It is clear that

Noting that the Ai's are mutually exclusive, we have

(b) Noting that B, 3 B, =, . . . 3 Bi 3 . . . , we have
w 00
3)
B,
U B~ = B, = {u: u I and 0 = {v: u r; 0)
i= 1 i= 1
1.1 1. Consider the switching networks shown in Fig. 1-5. Let A,, A,, and A, denote the events that
the switches s,, s,, and s, are closed, respectively. Let A,, denote the event that there is a closed
path between terminals a and b. Express A,, in terms of A,, A,, and A, for each of the networks
shown.

(4
(b)
Fig. 1-5

From Fig. 1-5(a), we see that there is a closed path between a and b only if all switches s,, s,, and s,
are closed. Thus,
A,, = A, n A, (3 A,
From Fig. 1-5(b), we see that there is a closed path between a and b if at least one switch is closed.
Thus,
A,, = A, u A, v A,
From Fig. 1-5(c), we see that there is a closed path between a and b if s, and either s, or s, are closed.
Thus,
A,, = A, n (A, v A,)
Using the distributive law (1.12), we have
A,, = (A1 n A,) u (A, n A,)
which indicates that there is a closed path between a and b if s, and s, or s, and s, are closed.
From Fig. 1-5(d), we see that there is a closed path between a and b if either s, and s, are closed or s,
is closed. Thus
A,, = (A, n A,) u A3