Page 12 - Schaum's Outlines - Probability, Random Variables And Random Processes

CHAP. 11 PROBABILITY

Distributive Laws:

De Morgan's Laws:

These relations are verified by showing that any element that is contained in the set on the left side of
the equality sign is also contained in the set on the right side, and vice versa. One way of showing this
is by means of a Venn diagram (Prob. 1.13). The distributive laws can be extended as follows:

Similarly, De Morgan's laws also can be extended as follows (Prob. 1.17):

1.4 THE NOTION AND AXIOMS OF PROBABILITY
An assignment of real numbers to the events defined in a sample space S is known as the prob-
ability measure. Consider a random experiment with a sample space S, and let A be a particular event
defined in S.

A. Relative Frequency Definition:
Suppose that the random experiment is repeated n times. If event A occurs n(A) times, then the
probability of event A, denoted P(A), is defined as

where n(A)/n is called the relative frequency of event A. Note that this limit may not exist, and in
addition, there are many situations in which the concepts of repeatability may not be valid. It is clear
that for any event A, the relative frequency of A will have the following properties:
1. 0 5 n(A)/n I 1, where n(A)/n = 0 if A occurs in none of the n repeated trials and n(A)/n = 1 if A
occurs in all of the n repeated trials.
2. If A and B are mutually exclusive events, then