1
Basic Probability Concepts
1.1
Introduction
1
1.2
Sample Space and Events
2
1.3
Definitions of Probability
4
1.4
Applications of Probability
7
1.5
Elementary Set Theory
8
1.6
Properties of Probability
13
1.7
Conditional Probability
15
1.8
Independent Events
26
1.9
Combined Experiments
29
1.10
Basic Combinatorial Analysis
31
1.11
Reliability Applications
42
1.12
Chapter Summary
47
1.13
Problems
47
1.14
References
57
1.1
Introduction
Probability deals with unpredictability and randomness, and probability theory
is the branch of mathematics that is concerned with the study of random phe
nomena. A random phenomenon is one that, under repeated observation, yields
different outcomes that are not deterministically predictable. However, these
outcomes obey certain conditions of statistical regularity whereby the relative
frequency of occurrence of the possible outcomes is approximately predictable.
Examples of these random phenomena include the number of electronic mail
(email) messages received by all employees of a company in one day, the num
ber of phone calls arriving at the university’s switchboard over a given period,
1
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Chapter 1
Basic Probability Concepts
the number of components of a system that fail within a given interval, and the
number of A’s that a student can receive in one academic year.
According to the preceding definition, the fundamental issue in random phe
nomena is the idea of a repeated experiment with a set of possible outcomes or
events. Associated with each of these events is a real number called the proba
bility of the event that is related to the frequency of occurrence of the event in a
long sequence of repeated trials of the experiment. In this way it becomes obvi
ous that the probability of an event is a value that lies between zero and one, and
the sum of the probabilities of the events for a particular experiment should sum
to one.
This chapter begins with events associated with a random experiment. Then it
provides different definitions of probability and considers elementary set theory
and algebra of sets. Finally, it discusses basic concepts in combinatorial analysis
that will be used in many of the later chapters.
1.2
Sample Space and Events
The concepts of
experiments
and
events
are very important in the study of prob
ability. In probability, an experiment is any process of trial and observation. An
experiment whose outcome is uncertain before it is performed is called a
random
experiment. When we perform a random experiment, the collection of possible
elementary outcomes is called the
sample space
of the experiment, which is usu
ally denoted by
S
. We define these outcomes as elementary outcomes because
exactly one of the outcomes occurs when the experiment is performed. The ele
mentary outcomes of an experiment are called the
sample points
of the sample
space and are denoted by
w
i
,
i
=
1
,
2
,...
. If there are
n
possible outcomes of an
experiment, then the sample space is
S
= {
w
1
,
w
2
,...,
w
n
}
.
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 Fall '07
 Carlton
 Set Theory, Conditional Probability, Probability, Probability theory, Naive set theory

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