Page 31 - Schaum's Outlines - Probability, Random Variables And Random Processes

PROBABILITY [CHAP 1

Verify that P(S) = 1.
The sample space of this experiment is

where e, is the elementary event that the first head appears on the kth toss.
Since a fair coin is tossed, we assume that a head and a tail are equally likely to appear. Then P(H) =
P(T) = $. Let

Since there are 2k equally likely ways of tossing a fair coin k times, only one of which consists of (k - 1)
tails following a head we observe that

Using the power series summation formula, we have

1.36. Consider the experiment of Prob. 1.35.
(a) Find the probability that the first head appears on an even-numbered toss.
(b) Find the probability that the first head appears on an odd-numbered toss.
(a) Let A be the event "the first head appears on an even-numbered toss." Then, by Eq. (1.36) and using
Eq. (1.79) of Prob. 1.35, we have

(b) Let B be the event "the first head appears on an odd-numbered toss." Then it is obvious that B = 2.
Then, by Eq. (1.25), we get

As a check, notice that

CONDITIONAL PROBABILITY
1.37. Show that P(A I B) defined by Eq. (1.39) satisfies the three axions of a probability, that is,
P(A(B) 2 0
P(S I B) = 1
P(A, u A, I B) = P(A, I B) + P(A, I B) if A, n A, = 0
From definition (1.39),

By axiom 1, P(A n B) 2 0. Thus,
P(AIB) 2 0