Page 309 -

error bars and the confidence level for any experimentally obtained result,
through a methodical analysis and reduction of the raw data.
In future courses in probability, random variables, stochastic processes
(which is random variables theory with time as a parameter), information
theory, and statistical physics, you will study techniques and solutions to the
different types of problems from the above list. In this very brief introduction
to the subject, we introduce only the very fundamental ideas and results —
where more advanced courses seem to almost always start.

10.2 Basics

Probability theory is best developed mathematically based on a set of axioms
from which a well-defined deductive theory can be constructed. This is
referred to as the axiomatic approach. We concentrate, in this section, on
developing the basics of probability theory, using a physical description of
the underlying concepts of probability and related simple examples, to lead
us intuitively to what is usually the starting point of the set theoretic axiom-
atic approach.
Assume that we conduct n independent trials under identical conditions,
in each of which, depending on chance, a particular event A of particular
interest either occurs or does not occur. Let n(A) be the number of experi-
ments in which A occurs. Then, the ratio n(A)/n, called the relative frequency
of the event A to occur in a series of experiments, clusters for n →∞ about
some constant. This constant is called the probability of the event A, and is
denoted by:

nA()
PA( ) = lim (10.1)
n→∞ n
From this definition, we know specifically what is meant by the statement
that the probability for obtaining a head in the flip of a fair coin is 1/2.
Let us consider the rolling of a single die as our prototype experiment :
1. The possible outcomes of this experiment are elements belonging
to the set:

S = {12 3 4 5 6,,, ,, } (10.2)

If the die is fair, the probability for each of the elementary elements
of this set to occur in the roll of a die is equal to:

1
P()1 = P( )2 = P( )3 = P( )4 = P( )5 = P( )6 = (10.3)
6

© 2001 by CRC Press LLC