2.1 Introduction
Probability theory and statistics forms the basis for the assessment of probabilities of
occurrence of uncertain events and thus constitutes a cornerstone in risk and decision analysis.
Only when a consistent basis has been established for the treatment of the uncertainties
influencing the probability that events with possible adverse consequences may occur it is
possible to assess the risks associated with a given activity and thus to establish a rational
basis for decision making.
Based on the probability theory and statistical assessments it is possible to represent the
uncertainties associated with a given engineering problem in the decision making process.
Aiming to provide a fundamental understanding of the notion of uncertainty this topic is
addressed with some detail. An appropriate representation of uncertainties is available through
probabilistic models such as random variables and random processes. The characterisation of
probabilistic models utilizes statistical information and the general principles for this are
finally shortly outlined.
2.2 Definition of Probability
The purpose of the theory of probability is to enable quantitative assessment of probabilities
but the real meaning and interpretation of probabilities and probabilistic calculations as such
is not a part of the theory. Consequently two people may have completely different
interpretations of the probability concept, but still use the same calculus. In the following,
three different interpretations of probability are introduced and discussed based on simple
cases. A formal presentation of basic set theory together with the axioms of probability theory
may be found in the lecture notes on Basic Theory of Probability and Statistics in Civil
Engineering (Faber, 2006).
Frequentistic Definition
The frequentistic definition of probability is the typical interpretation of probability of the
experimentalist. In this interpretation the probability is simply the relative frequency of
occurrence of the event
PA()
A as observed in an experiment with n trials, i.e. the probability of
an event is defined as the number of times that the event occurs divided by the number
of experiments that is carried out:
AA
exp
exp( )=lim A forN
PA n
n(2.1)
where:
NA= number of experiments where A occurred
nexp= total number of experiments.If a frequentist is asked what is the probability for achieving a “head” when flipping a coin
she would principally not know what to answer until she would have performed a large
number of experiments. If say after 1000 experiments (flips with the coin) it is observed that