2.6 Brief introduction to probabilities

The topic of probability is popularly associated with gambling and weather forecasts. Simple probability problems are indeed related to dice-rolling and coin-tossing predictions and composers have always been interested in using the die for composing music; Mozart's musical dice game for composing waltzes comes to mind here (Schwanauer and Levitt, 1993).

As an example, consider the prediction for scoring more than four with a single throw of a die. Technically speaking, the throw of a die constitutes a trial. A trial is a process which can be expected to produce one outcome from the set of all possible outcomes; these outcomes cannot occur simultaneously. In the case of the die, the set of all possible outcomes from the dierolling is S = {1, 2, 3, 4, 5, 6}. Our task is to predict when the result will be either five or six; this is normally referred to as the set of events E = {5, 6}. A further aspect to consider in a prediction is whether the trial is fair. That is, whether or not the probability of obtaining any one outcome is the same as for all other outcomes. In the case of the die, the trial is fair but there are many cases in which some outcomes are favoured over others (this will be discussed further in Chapter 3).

The chances of obtaining either the number five or six are two in six: the ratio of the number of...