Page 30 - Schaum's Outlines - Probability, Random Variables And Random Processes

CHAP. 11 PROBABILITY 2 3

Consider a sample space S of which a typical outcome is (1,0,0, I), indicating that switches 1 and 4 are
closed and switches 2 and 3 are open. The sample space contains 24 = 16 points, and by assumption, they
are equally likely (Fig. 1-13).
Let A,, i = 1, 2, 3, 4 be the event that the switch si is closed. Let A be the event that there exists a
closed path between a and b. Then
A = A, u (A, n A,) u (A2 n A,)
Applying Eq. (1 JO), we have

Now, for example, the event A, n A, contains all elementary events with a 1 in the second and third
places. Thus, from Fig. 1-13, we see that
n(A,) = 8 n(A2 n A,) = 4 n(A2 n A4) = 4
n(A, n A, n A,) = 2 n(A, n A, n A,) = 2
n(A, n A, n A,) = 2 n(A, n A, n A, n A,) = 1
Thus,

Fig. 1-13

1.35. Consider the experiment of tossing a fair coin repeatedly and counting the number of tosses
required until the first head appears.
(a) Find the sample space of the experiment.
(b) Find the probability that the first head appears on the kth toss.