30

Probability

All scientists will know the importance of experiment and observation and,

equally, be aware that the results of some experiments depend to a degree on

chance. For example, in an experiment to measure the heights of a random sample

of people, we would not be in the least surprised if all the heights were found to

be different; but, if the experiment were repeated often enough, we would expect

to find some sort of regularity in the results. Statistics, which is the subject of the

next chapter, is concerned with the analysis of real experimental data of this sort.

First, however, we discuss probability. To a pure mathematician, probability is an

entirely theoretical subject based on axioms. Although this axiomatic approach is

important, and we discuss it briefly, an approach to probability more in keeping

with its eventual applications in statistics is adopted here.

We first discuss the terminology required, with particular reference to the

convenient graphical representation of experimental results as Venn diagrams.

The concepts of random variables and distributions of random variables are then

introduced. It is here that the connection with statistics is made; we assert that

the results of many experiments are random variables and that those results have

some sort of regularity, which is represented by a distribution. Precise definitions

of a random variable and a distribution are then given, as are the defining

equations for some important distributions. We also derive some useful quantities

associated with these distributions.

30.1 Venn diagrams

We call a single performance of an experiment atrialand each possible result

anoutcome.Thesample spaceSof the experiment is then the set of all possible

outcomes of an individual trial. For example, if we throw a six-sided die then there

are six possible outcomes that together form the sample space of the experiment.

At this stage we are not concerned with how likely a particular outcome might