Page 30 - Fundamentals of Probability and Statistics for Engineers

Basic Probability Concepts 13

Table 2.1 Corresponding statements in set theory and probability
Set theory Probability theory
Space, S Sample space, sure event
Empty set, ; Impossible event
Elements a, b, ... Sample points a, b, ... or simple events)
Sets A, B, .. . Events A, B, .. .
A Event A occurs
A Event A does not occur
A [ B At least one of A and B occurs
AB Both A and B occur
A B A is a subevent of B i.e. the occurrence of A necessarily implies
the occurrence of B)
AB ˆ; A and B are mutually exclusive i.e. they cannot occur
simultaneously)

A,B,C,.... Some of these corresponding sets and probability meanings are
given in Table 2.1. As Table 2.1 shows, the empty set ; is considered an
impossible event since no possible outcome is an element of the empty set.
Also, by ‘occurrence of an event’ we mean that the observed outcome is an
element of that set. For example, event A [ B is said to occur if and only if the
observed outcome is an element of A or B or both.
Example 2.4. Consider an experiment of counting the number of left-turning
cars at an intersection in a group of 100 cars. The possible outcomes (possible
numbers of left-turning cars) are 0, 1, 2, ... , 100. Then, the sample space S is
S ˆf0, 1, 2, ... , 100g . Each element of S is a sample point or a possible out-
come. The subset A ˆf0, 1, 2, ... ,50g is the event that there are 50 or fewer
cars turning left. The subset B ˆf40, 41, ... ,60g is the event that between 40
and 60 (inclusive) cars take left turns. The set A [ B is the event of 60 or fewer
cars turning left. The set A \ B is the event that the number of left-turning cars
is between 40 and 50 (inclusive). Let C ˆf80, 81, ... , 100g Events A and C are
mutually exclusive since they cannot occur simultaneously. Hence, disjoint sets
are mutually exclusive events in probability theory.

2.2.1 AXIOMS OF PROBABILITY

We now introduce the notion of a probability function. Given a random experi-
ment, a finite number P(A) is assigned to every event A in the sample space S of
all possible events. The number P(A) is a function of set A and is assumed to
be defined for all sets in S. It is thus a set function, and P(A) is called the
probability measure of A or simply the probability of A. It is assumed to have the
following properties (axioms of probability):

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