Page 44 - Probability, Random Variables and Random Processes

36 PROBABILITY [CHAP 1

Let A and B be any two events in S. Express the following events in terms of A and B.
(a) At least one of the events occurs.
(b) Exactly one of two events occurs.
Ans. (a) A u B; (b) A A B

Let A, B, and C be any three events in S. Express the following events in terms of these events.
(a) Either B or C occurs, but not A.
(b) Exactly one of the events occurs.
(c) Exactly two of the events occur.
Ans. (a) A n (B v C)
(b) (A n (B u C)) u (B n (A u C)) u (C n (A u B))
(c) ((A n B) n C) u {(A n C) n B) u {(B n C) n A)

A random experiment has sample space S = {a, h, c). Suppose that P({a, c)) = 0.75 and P({b, c)) = 0.6.
Find the probabilities of the elementary events.
Ans. P(a) = 0.4, P(b) = 0.25, P(c) = 0.35

Show that
(a) P(A u B) = 1 - P(A n B)
(b) P(A n B) 2 1 - P(A) - P(B)

Hint: (a) Use Eqs. (1.1 5) and (1 .Z).
(b) Use Eqs. (1 .29), (l.25), and (1.28).
(c) See Prob. 1.68 and use axiom 3.

Let A, B, and C be three events in S. If P(A) = P(B) = 4, P(C) = 4, P(A n B) = 4, P(A n C) = 6, and
P(B n C) = 0, find P(A u B u C).
Ans. 2

Verify Eq. (1.30).
Hint: Prove by induction.

Show that
P(A, n A, n - - n A,) 2 P(A,) + P(AJ + . - + P(A,) - (n - 1)
Hint: Use induction to generalize Bonferroni's inequality (1.63) (Prob. 1.22).
In an experiment consisting of 10 throws of a pair of fair dice, find the probability of the event that at least
one double 6 occurs.
Ans. 0.246

Show that if P(A) > P(B), then P(A I B) > P(B I A).
Hint : Use Eqs. (1.39) and (1 .do).

An urn contains 8 white balls and 4 red balls. The experiment consists of drawing 2 balls from the urn
without replacement. Find the probability that both balls drawn are white.
Ans. 0.424