Risk Assessment and Decision Analysis with Bayesian Networks

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4

The Basics of Probability

4.1 Introduction

In discussing the difference between the frequentist and subjective

approaches to measuring uncertainty, we were careful in Chapter 3 not

to mention the word probability. That is because we want to dene prob-

ability in such a way that it makes sense for whatever reasonable approach

to measuring uncertainty we choose, be it frequentist, subjective, or even

an approach that nobody has yet thought of. To do this in Section 4.2 we

describe some properties (called axioms) that any reasonable measure of

uncertainty should satisfy; then we dene probability as any measure

that satises those properties. The nice thing about this way of dening

probability is that not only does it avoid the problem of vagueness, but it

also means that we can have more than one measure of probability. In

particular, we will see that both the frequentist and subjective approaches

satisfy the axioms, and hence both are valid ways of dening

probability.

In Section 4.3 we introduce the crucial notion of probability distri-

butions. In Section 4.4 we use the axioms to dene the crucial issue

of independence of events. An especially important probability dis-

tribution—the Binomial distribution—which is based on the idea of

independent events, is described in Section 4.5. Finally in Section 4.6

we will apply the lessons learned in the chapter to solve some of the

problems we set in Chapter 1 and debunk a number of other probability

fallacies.

4.2 Some Observations Leading to Axioms

and Theorems of Probability

Before stating the axioms of probability we are going to list some points

that seem to be reasonable and intuitive for both the frequentist and

subjective denitions of chance. So, consider again statements like the

following:

◾ There is a 50% chance of tossing a head on a fair coin.

◾ There is a 0.00001% chance of a meteor destroying the White

House in the next 5 years.

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