The Maths of Probability

Mathematical results and application formulae start from axioms. When learning maths, it is wise to develop solutions from the axioms until the process becomes familiar and then we see the application of the formulae as being straightforward. Also along this road, we get a feel for what the maths mean and can relate it to more concrete experiences.

Calculations in probability are almost trivial but defining the problem, that is the analysis, is non-trivial. If decoding of the problem into the appropriate mathematical form is not obvious, a sound strategy is to construct the analysis from the underlying principles, ie. the axioms.

In this section, we start with the axioms of probability which must be learnt so that they can be readily recalled. With this advantage, we can interpret the problem.

The maths of probability is founded in Set Theory and before we express the ideas with mathematical symbols, we require some definitions.

• A random experiment is characterized by:
  1. It can be repeated indefinitely under essentially unchanged conditions.
  2. Although we cannot state what a particular outcome will be, we are able to list all possible outcomes.
  3. As the experiment is performed repeatedly, individual outcomes occur in a haphazard fashion, but after many repetitions a pattern of regularity may appear.
  An analogy of point 3 is the distribution of drops from a paint spray can. Although we cannot reliably predict where each drop will hit the surface, after a short burst we can identify the spray pattern.

• The Sample space is the set of all possible outcomes and each point in the sample space is termed a sample point.

• An Event is a collection of sample points and is a subset of of the sample space. In maths we write,
  $$\begin{eqnarray*} S & = & \{\mbox{all possible outcomes} \} \\ E & = & \{\mbo... ...ubseteq & S \mbox{(This reads \lq\lq E is a subset of S''.)}\\ \end{eqnarray*}$$
  
  
  
  
  

  The complement of the event, denoted by $\bar E$, is the part of $S$ which is not in $E$.

Figure 2 shows a Venn diagram of a Sample space, an Event and the complement of the Event.

Figure 2: Sample and Event spaces