In this appendix we review some basic concepts and results of probability theory that will be of use in this book. It is meant to be a refresher of concepts that the reader has already seen elsewhere in a more complete form and at the same time as a way to establish notation. The choice of topics reflects the emphasis of this book on using mean values, variances, and covariances. Useful textbooks, which could be used in a formal course on probability theory, are listed at the end of this appendix.

Introduction

Experiments

Probability theory is concerned with “experiments” with multiple possible outcomes. An example of an experiment is to toss a coin. The only outcomes (also known as realizations or sample values) are “heads” and “tails.” We say that the ensemble of all possible realizations of this experiment are heads and tails. Another example, with many possible outcomes, is to spin a wheel of fortune.

Consider now the problem of predicting the piezometric head of an aquifer at a location. If we cannot predict the outcome but we realize that there are many possible outcomes (predictions), then it is useful to conceptualize the process the same way we do a game of chance. Prediction with incomplete information and games of chance share the characteristic that their outcomes cannot be predicted with certainty and we have to come up with a different (from the usual deterministic) way to make predictions.