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In axiomatic probability, t he term event has special meaning and should not
be used interchangeably with outcome. An event is a special subset of the sample
space S. We usually wish to consider various events defined on a sample space,
and they will be denoted with uppercase letters such as A, B, C,...,orperhaps
A
1
, A
2
, . . . , etc. Also, we will have occasion to consider the set of operations of
union, intersection, and complement of our de fined events. Thus, we must be
careful in our definition of events to make the set sufficiently complete such that
these set operations also yield properly defined events. In discrete problems, this
can always be done by defining the set of events under consideration to be all
possible subsets of the sample space S. We will tacitly assume that the null set is a
subset of every set, and that every set is a subset of itself.
One other comment about events is in order before proceeding to the basic
axioms of probability. The event A is said to occur if any point in A occurs.
The three axioms of probability may now be stated. Let S be the sample space
and A be any event defined on the sample space. The first two axioms are
Axiom 1: PðAÞ0 (1.3.1)
Axiom 2: PðSÞ¼1 (1.3.2)
Now, let A
1
, A
2
, A
3
, . . . be mutually exclusive (disjoint) events defined on S. The
sequence may be finite or countably infinite. The third axiom is then
Axiom 3: PðA
1
[ A
2
[ A
3
[ ...Þ
¼ PðA
1
ÞþPðA
2
ÞþPðA
3
Þþ
(1.3.3)
Axiom 1 simply says that the probability of an event cannot be negative. This
certainly conforms to the relative-frequency-of-occurrence concept of probability.
Axiom 2 says that the event S, which includes all possible outcomes, must have a
probability of unity. It is sometimes called the certain event. The first two axioms
are obviously necessary if axiomatic probability is to be compatible with the older
relative-frequency probability theory. The third axiom is not quite so obvious,
perhaps, and it simply must be assumed. In words, it says that when we have
nonoverlapping (disjoint) events, the probability of the union of these events is the
sum of the probabilities of the individual events. If this were not so, one could easily
think of counterexamples that would not be compatible with the relative-frequency
concept. This would be most undesirable.
We now recapitulate. There are three essential ingredients in the formal
approach to probability. First, a sample space must be defined that includes all
possible outcomes of our conceptual experiment. We have some discretion in what
we call outcomes, but caution is in order here. The outcomes must be disjoint and
all-inclusive such that P(S) ¼1. Second, we must carefully define a set of events on
the sample space, and the set must be closed such that me operations of union,
intersection, and complement also yield events in the set. Finally, we must assign
probabilities to all events in accordance with the basic axioms of probability. In
physical problems, this assignment is chosen to be compatible with what we feel to
be reasonable in terms of relative frequency of occurrence of the events. If the
sample space S contains a finite number of elements, the probability assignment is
usually made directly on the elements of S. They are, of course, elementary events
6 CHAPTER 1 PROBABILITY AND RANDOM VARIABLES: A REVIEW
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