Page 19 - Probability, Random Variables and Random Processes

CHAP. 1) PROBABILITY

A
Fig. 1-3

1.6. An automobile dealer offers vehicles with the following options:
(a) With or without automatic transmission
(b) With or without air-conditioning
(c) With one of two choices of a stereo system
(d) With one of three exterior colors ,
If the sample space consists of the set of all possible vehicle types, what is the number of out-
comes in the sample space?
The tree diagram for the different types of vehicles is shown in Fig. 1-4. From Fig. 1-4 we see that the
number of sample points in S is 2 x 2 x 2 x 3 = 24.

Transmission Automatic

Air-conditioning

Stereo

Color

Fig. 1-4

1.7. State every possible event in the sample space S = {a, b, c, d).
There are z4 = 16 possible events in S. They are 0; {a), (b), {c), {d) ; {a, b), {a, c), {a, d), {b, c),
{b, d), (c, d) ; {a, b, c), (a, b, 4, (a, c, d), {b, c, d) ; S = {a, b, c, dl-

1.8. How many events are there in a sample space S with n elementary events?
Let S = {s,, s,, . . . , s,). Let Q be the family of all subsets of S. (a is sometimes referred to as the power
set of S.) Let Si be the set consisting of two statements, that is,
Si = (Yes, the si is in; No, the s, is not in)
Then Cl can be represented as the Cartesian product
n = s, x s, x ... x s,
= ((s,, s2, . . . , s,): si E Si for i = 1, 2, . . . , n)