Page 30 - Probability, Random Variables and Random Processes

22 PROBABILITY [CHAP 1

Let B be the event that at least two persons have the same birthday. Then B = 2 and by Eq. (1.25),
P(B) = 1 - P(A).
(b) Substituting n = 50 in Eq. (1.78), we have
P(A) z 0.03 and P(B) z 1 - 0.03 = 0.97
(c) From Eq. (1.78), when n = 23, we have
P(A) x 0.493 and P(B) = 1 - P(A) w 0.507
That is, if there are 23 persons in a room, the probability that at least two of them have the same
birthday exceeds 0.5.

1.33. A committee of 5 persons is to be selected randomly from a group of 5 men and 10 women.
(a) Find the probability that the committee consists of 2 men and 3 women.
(b) Find the probability that the committee consists of all women.
(a) The number of total outcomes is given by

It is assumed that "random selection" means that each of the outcomes is equally likely. Let A be the
event that the committee consists of 2 men and 3 women. Then the number of outcomes belonging to
A is given by

Thus, by Eq. (l.38),

(b) Let B be the event that the committee consists of all women. Then the number of outcomes belonging
to B is

Thus, by Eq. (l.38),

1.34. Consider the switching network shown in Fig. 1-12. It is equally likely that a switch will or
will not work. Find the probability that a closed path will exist between terminals a and b.

Fig. 1-12