In this section probability theory and random variables are introduced.
Key results that underpin further sections are given here.
If more information is required, the book *Probability and Random Processes*
by Grimmett & Stirzaker (1982) is a good place to start.

Suppose a needle is dropped onto a floor made up of planks of wood.
The needle may or may not intersect one of the joints between the planks.
A single throw of the needle is called an *experiment* or *trial*.
There are two possible *outcomes*. Either the needle intersects a joint(s)
or it lands between the joints. By repeating the experiment a large number of times,
the probability \(\mathbf{P}\) of a particular *outcome* or *event* can be
calculated. Let \(A\) be the event "needle intersects a joint".
Let \(N(A)\) be the number of occurrences of \(A\) over \(n\) trials.
As \(n\to\infty\), \(N(A)/n\) converges to the probability that \(A\) occurs,
\(\mathbf{P}(A)\), on any particular trial. On occasions, such as
Buffon's needle throwing experiment (1777),
a probability can be assigned to an outcome without experiment.
The set of all possible outcomes of an experiment is called the *sample space*
and is denoted by \(\Omega\). A *random variable* is a function \(X : \Omega\to\real\).

Uppercase letters will be used to represent generic random variables, whilst lowercase
letters will be used to represent possible numerical values of these variables.
To describe the probability of possible values of \(X\), consider the following
definition. The *distribution function* of a random variable \(X\) is the
function \(F_{X} : \real \to [0, 1]\) given by \(F_{X}(x) = \mathbf{P}(X \le x)\).

BUFFON, G. L. L. Comte de. Essai d'Arithmétique Morale. In: Supplément à l'Histoire Naturelle, v. 4. Paris: Imprimerie Royale (1777).

GRIMMETT, G. and STIRZAKER, D. Probability and Random Processes, Clarendon Press, Oxford (1982).