Page 40 - Probability, Random Variables and Random Processes

PROBABILITY [CHAP 1

We have
P[A n (B u C)] = P[(A n B) u (A n C)] [Eq. (1.12)1
=P(AnB)+P(AnC)-P(AnBnC) [Eq.(1.29)]
= P(A)P(B) + P(A)P(C) - P(A)P(B)P(C) CEq. (1WI
= P(A)P(B) + P(A)P(C) - P(A)P(B n C) [Eq. (1.50)]
= P(A)[P(B) + P(C) - P(B n C)]
= P(A)P(B u C) CE~. (1.2911
Thus, A and (B u C) are independent.

1.56. Consider the experiment of throwing two fair dice (Prob. 1.31). Let A be the event that the sum
of the dice is 7, B be the event that the sum of the dice is 6, and C be the event that the first die is
4. Show that events A and C are independent, but events B and C are not independent.
From Fig. 1-3 (Prob. l.5), we see that

and
Now
and
Thus, events A and C are independent. But

Thus, events B and C are not independent.

1.57. In the experiment of throwing two fair dice, let A be the event that the first die is odd, B be the
event that the second die is odd, and C be the event that the sum is odd. Show that events A, B,
and C are pairwise independent, but A, B, and C are not independent.
From Fig. 1-3 (Prob. 1.5), we see that

Thus

which indicates that A, B, and C are pairwise independent. However, since the sum of two odd numbers is
even, (A n B n C) = 0 and
P(A n B n C) = 0 # $ = P(A)P(B)P(C)
which shows that A, B, and C are not independent.

1.58. A system consisting of n separate components is said to be a series system if it functions when all
n components function (Fig. 1-16). Assume that the components fail independently and that the
probability of failure of component i is pi, i = 1, 2, . . . , n. Find the probability that the system
functions.

Fig. 1-16 Series system.