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How to think about probability

The approach to the study of chance in earlier years has been quite intuitive. This is a reasonable approach which is sufficient for many purposes; many people who have not studied probability in detail manage to think about chance events quite correctly. For example, some professional gamblers know about odds and can manage risk in a sensible way, even making an income, without necessarily having a strong mathematical background.

However, this intuitive approach only goes so far. And this is the point in the curriculum at which the consideration of probability is formalised. In order to do this, we start by thinking in a fundamental way about probability statements.

The Bureau of Meteorology may say, for example, that the probability of rain tomorrow is 20%, or 0.20 on a scale of 0 to 1. We are immediately reminded that a probability is a number between 0 and 1 (inclusive), or 0% and 100% (inclusive) if the percentage scale is used.

These are hardly different scales, they simply provide two different ways to express the same number. Nonetheless, because both usages are common, it is important to recognise in a given context whether probabilities are being expressed as numbers between 0 and 1, or percentages between 0% and 100%. There is potential for confusion, especially where very small probabilities are concerned.

So if a probability is a number, how is it obtained, or defined? A probability does not come from an ordinary mathematical operation in the usual sense of functions, such as logarithms and the trigonometric functions. When we speak in a natural way about chance and probability, we refer to something that may happen (an event), and we assign a number to the probability of the event.

This sounds a bit like a functional statement, which is an important insight. When we write \(f(x) = 2\), we say `\(f\) of \(x\) equals 2'. By analogy, we say `the probability of rain tomorrow equals 0.2'.

From this perspective, we can see that the domain of probability statements consists of events: things that may happen. But — to state the obvious — there are many things that may happen, and not only that, there is a plethora of types of events in a wide variety of contexts. Ordinary mathematical functions — at any rate, the ones students have met so far — tend to have simple numerical domains such as \(\mathbb{R}\) or \(\mathbb{R}^+\).

We need a different mathematical structure to deal with probability, which we discuss in this module.

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