Probability: What does it mean?

(a)

When we calculate a probability, is that merely an approximation method forced on us by our incomplete information?

(b)

Is probability merely a set of mathematical rules for expressing degrees of uncertainty?

(c)

Is the calculation of the probability that a certain event will occur a complete description of the event?

The scientific method requires that theories be validated by experiments and data. By experiment we mean the researcher measuring some process which may be (i) controlled or (ii) uncontrolled. An example of a controlled experiment is running motors under different stresses to measure their lifetimes. Measurement of weather variables is an uncontrolled experiment.

Since the outcomes will include uncertainty, we require mathematics to convey the meaning of that uncertainty and this sector of mathematics is called probability. Whilst this talk is restricted to probability, mathematical analysis blends many mathematical ideas (eg calculus, geometry, algebra) and in what may seem a paradox, the more general or abstract the maths becomes, the more applicable it is. The distinction between pure and applied maths is at best fuzzy, if indeed such a boundary even exists.

The concept of probability is not foreign. If the weather report predicts rain with probability of $0.6$ (say), then we interpret this as:

(a)

on days when similar atmospheric conditions have been measured, it has rained on $60\%$ of them, or

(b)

0.6 is a measure of the forecasters belief that it will rain.

In (a), the probability of the outcome is regarded as being a property of that outcome. It is imagined that this property can be determined by continual repetition of observing weather and the probability of the outcome will be the proportion of observations that result in that outcome.

In (b), the probability of outcome is a statement about the beliefs of the person concerning the chance that the outcome will occur.

Regardless of which interpretation is adopted, the mathematics of probability is the same.