Page 29 - Fundamentals of Probability and Statistics for Engineers

12 Fundamentals of Probability and Statistics for Engineers

The second relation in Equations (2.10) gives the union of two sets in terms
of the union of two disjoint sets. As we will see, this representation is useful in
probability calculations. The last two relations in Equations (2.10) are referred
to as DeMorgan’s laws.

2.2 SAMPLE SPACE AND PROBABILITY MEASURE

In probability theory, we are concerned with an experiment with an outcome
depending on chance, which is called a random experiment. It is assumed that all
possible distinct outcomes of a random experiment are known and that they are
elements of a fundamental set known as the sample space. Each possible out-
come is called a sample point, and an event is generally referred to as a subset of
the sample space having one or more sample points as its elements.
It is important to point out that, for a given random experiment, the
associated sample space is not unique and its construction depends upon the
point of view adopted as well as the questions to be answered. For example,
100
resistors are being manufactured by an industrial firm. Their values,
owing to inherent inaccuracies in the manufacturing and measurement pro-

cesses, may range from 99 to 101 . A measurement taken of a resistor is a
random experiment for which the possible outcomes can be defined in a variety
of ways depending upon the purpose for performing such an experiment. On
the one hand, if, for a given user, a resistor with resistance range of 99.9–100.1
is considered acceptable, and unacceptable otherwise, it is adequate to define
the sample space as one consisting of two elements: ‘acceptable’ and ‘unaccept-
able’. On the other hand, from the viewpoint of another user, possible

outcomes may be the ranges 99–99 5 :
, 99.5–100 , 100–100.5 , and

100.5–101 . The sample space in this case has four sample points. Finally, if
each possible reading is a possible outcome, the sample space is now a real line
from 99 to 101 on the ohm scale; there is an uncountably infinite number of
sample points, and the sample space is a nonenumerable set.
To illustrate that a sample space is not fixed by the action of performing the
experiment but by the point of view adopted by the observer, consider an
energy negotiation between the United States and another country. From the
point of view of the US government, success and failure may be looked on as
the only possible outcomes. To the consumer, however, a set of more direct
possible outcomes may consist of price increases and decreases for gasoline
purchases.
The description of sample space, sample points, and events shows that they
fit nicely into the framework of set theory, a framework within which the
analysis of outcomes of a random experiment can be performed. All relations
between outcomes or events in probability theory can be described by sets and
set operations. Consider a space S of elements a, b, c,..., and with subsets

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