Page 32 - Schaum's Outlines - Probability, Random Variables And Random Processes

CHAP. 11 PROBABILITY

(b) By Eq. (IS), S n B = B. Then

(c) By definition (1.39),

Now by Eqs. (1.8) and (1.1 I), we have
(A, u A,) n B=(Al n B) u (A, n B)
and A, n A, = 0 implies that (A, n B) n (A, n B) = 0. Thus, by axiom 3 we get

1.38. Find P(A I B) if (a) A n B = a, (b) A c B, and (c) B c A.
then
(a) If A n B = 0, P(A n B) = P(0) = 0. Thus,

(b) If A c B, then A n B = A and

(c) If B c A, then An B= Band

1.39. Show that if P(A B) > P(A), then P(B I A) > P(B).
P(A n B)
If P(A I B) = -------- > P(A), then P(A n B) > P(A)P(B). Thus,
P(B)

1.40. Consider the experiment of throwing the two fair dice of Prob. 1.31 behind you; you are then
informed that the sum is not greater than 3.

(a) Find the probability of the event that two faces are the same without the information given.
(b) Find the probability of the same event with the information given.

(a) Let A be the event that two faces are the same. Then from Fig. 1-3 (Prob. 1.5) and by Eq. (1.38), we
have
A = {(i, i): i = 1, 2, ..., 6)
and