Page 29 - Probability, Random Variables and Random Processes

CHAP. 11 PROBABILITY

* From the observation, we have
P(dot) = 2P(dash)
Then, by Eq. (1.39,
P(dot) + P(dash) = 3P(dash) = 1
Thus, P(dash) = 5 and P(dot) =

1.30. The sample space S of a random experiment is given by
S = {a, b, c, d]
with probabilities P(a) = 0.2, P(b) = 0.3, P(c) = 0.4, and P(d) = 0.1. Let A denote the event {a, b),
and B the event {b, c, d). Determine the following probabilities: (a) P(A); (b) P(B); (c) P(A); (d)
P(A u B); and (e) P(A n B).
Using Eq. (1.36), we obtain

(a) P(A) = P(u) + P(b) = 0.2 + 0.3 = 0.5
(b) P(B) = P(b) + P(c) + P(d) = 0.3 + 0.4 + 0.1 = 0.8
(c) A = (c, d); P(4 = P(c) + P(d) = 0.4 + 0.1 = 0.5
(d) A u B = {a, b, c, d) = S; P(A u B) = P(S) = 1
(e) A n B=(b};P(A n B)= P(b)=O.3

1.31. An experiment consists of observing the sum of the dice when two fair dice are thrown (Prob.
1.5). Find (a) the probability that the sum is 7 and (b) the probability that the sum is greater than

Let rij denote the elementary event (sampling point) consisting of the following outcome: cij = (i, j),
where i represents the number appearing on one die and j represents the number appearing on the
other die. Since the dice are fair, all the outcomes are equally likely. So P(rij) = &. Let A denote the
event that the sum is 7. Since the events rij are mutually exclusive and from Fig. 1-3 (Prob. IS), we
have
P(A) = K16 u (25 u (34 u C43 u (52 u (6,)
= p(r16) + P(C25) + p(c34) + P(c421) + p(C52) + p(661)
= 6(&) = 4
Let B denote the event that the sum is greater than 10. Then from Fig. 1-3, we obtain
P(B) = P(556 u c65 u (66) = PG6) -1 P(C65) + W66)
= 3(&) =

1.32. There are n persons in a room.
(a) What is the probability that at least two persons have the same birthday?
(b) Calculate this probability for n = 50.
(c) How large need n be for this probability to be greater than 0.5?
(a) As each person can have his or her birthday on any one of 365 days (ignoring the possibility of
February 29), there are a total of (365)" possible outcomes. Let A be the event that no two persons
have the same birthday. Then the number of outcomes belonging to A is

Assuming that each outcome is equally likely, then by Eq. (1.38),