Introduction

The probability theory, and this one is no exception. However, the careful reader should wonder title mentions probability theories. In Section 5.1 we show that probability, like uncertainty, is a rather elusive concept. Descriptive statistics suggests the concept of probabilities as relative frequencies, but we may also interpret probability as plausibility related to a state of belief. The origin of the mathematical approach to probability can be traced back to Jacob Bernoulli, Thomas Bayes, and Pierre-Simon Laplace. Bernoulli’s Ars Conjectandi (The Art of Conjecture) was published 8 years after his death in 1713, and Laplace published his Théorie analytique des probabilités in 1812. More recently, the axiomatic approach due to Andrei Nikolaevich Kolmogorov (1933) was proposed and has become a sort of standard approach to probability. We will follow the last approach in this and subsequent, because it suits our purpose very well, but it is always healthy to keep in mind that “standard” does not mean “always the best.” We come back to such issues while we first introduce the axiomatic approach to probability theory in Section 5.2, laying down the fundamental concepts of events and probability measures, along with a set of basic rules of the game in order to work with probabilities in a sensible and consistent manner. In Section 5.3 we introduce conditional probabilities; we do so in a mathematically unsophisticated way, but we insist that conditioning is a powerful and essential concept that is used to model information availability to decision makers. Conditional probabilities also lead to a powerful result called Bayes’ theorem, which we will come to appreciate in Section 5.4.

We keep mathematical sophistication to a minimum, since our purpose is just to introduce the essential concepts that are used later to study random variables and inferential statistics. Some advanced topics in probability do require a more in-depth treatment based on a more sophisticated mathematical machinery. We provide references for the interested reader, and we will just give a little flavor of this later, in Section 7.10, in the context of random variables.