This chapter deals with a special subject and may be omitted on a first reading. Its contents are important and useful, but are not a prerequisite for most of the following chapters.

First Principles

We have seen that many interesting problems in probability can be solved by counting the number of outcomes in an event. Such counting often turns out to also be useful in more general contexts. This chapter sets out some simple methods of dealing with the commonest counting problems.

The basic principles are pleasingly easy and are perfectly illustrated in the following examples.

(1) Principle If I have m garden forks and n fish forks, then I have m+ n forks altogether.

(2) Principle If I have m different knives and n different forks, then there are mn distinct ways of taking a knife and fork.

These principles can be rephrased in general terms involving objects, operations, or symbols and their properties, but the idea is already obvious. The important points are that in (1), the two sets in question are disjoint; that is a fork cannot be both a garden fork and a fish fork. In (2), my choice of knife in no way alters my freedom to choose any fork (and vice versa).

Real problems involve, for example, catching different varieties of fish, drawing various balls from a number of urns, and dealing hands at numerous types of card games. In the standard terminology for such problems, we say that a number n(say) of objects or things are to be divided or distributed into r classes or groups.

This chapter introduces the basic concepts of probability in an informal way. We discuss our everyday experience of chance, and explain why we need a theory and how we start to construct one. Mathematical probability is motivated by our intuitive ideas about likelihood as a proportion in many practical instances. We discuss some of the more common questions and problems in probability, and conclude with a brief account of the history of the subject.

Chance

You can be reasonably confident that the sun will rise tomorrow, but what it will be shining on is a good deal more problematical. In fact, the one thing we can be certain of is that uncertainty and randomness are unavoidable aspects of our experience.

At a personal level, minor ailments and diseases appear unpredictably and are resolved not much more predictably. Your income and spending are subject to erratic strokes of good or bad fortune.

This book provides an introduction to elementary probability and some of its simple applications. In particular, a principal purpose of the book is to help the student to solve problems. Probability is now being taught to an ever wider audience, not all of whom can be assumed to have a high level of problem-solving skills and mathematical background. It is also characteristic of probability that, even at an elementary level, few problems are entirely routine. Successful problem solving requires flexibility and imagination on the part of the student. Commonly, these skills are developed by observation of examples and practice at exercises, both of which this text aims to supply.

With these targets in mind, in each chapter of the book, the theoretical exposition is accompanied by a large number of examples and is followed by worked examples incorporating a cluster of exercises.

The geometric and dynamic theory of the limit set generated by the iteration of finitely many similarity maps satisfying the open set condition has been well developed for some time now. Over the past several years, the authors have in turn developed a technically more complicated geometric and dynamic theory of the limit set generated by the iteration of infinitely many uniformly contracting conformal maps, a (hyperbolic) conformal iterated function system. This theory allows one to analyze many more limit sets, for example sets of continued fractions with restricted entries. We recall and extend this theory in the later chapters. The main focus of this book is the exploration of the geometric and dynamic properties of a far reaching generalization of a conformal iterated function system called a graph directed Markov system (GDMS). These systems are very robust in that they apply to many settings that do not fit into the scheme of conformal iterated systems. While the basic theory is laid out here and we touch on many natural questions arising in its context, we emphasize that there are many issues and current research topics which we do not cover: for examples, the detailed analysis of the structure of harmonic measures of limit sets provided in [UZd], the examination of the doubling property of conformal measures performed in [MU6], the extensive study of generalized polynomial like mappings (see [U7] and [SU]), the multifractal analysis of geometrically finite Kleinian groups (see [KS]), and the connection to quantization dimension from engineering (see [LM] and [GL]). There are many research problems in this active area that remain unsolved.