Applications in science, medicine, and operations research
We may assess or interpret probabilities in different ways according to the context. But, as David Hand wrote in his Statistics: A Very Short Introduction,‘... the calculus is the same’, i.e. how probabilities are manipulated does not change.
Keep in mind the central ideas of the subject: the Addition and Multiplication Laws; independence; the Laws of Large Numbers linking frequencies to objective probabilities; Gaussian distributions when summing random quantities; other frequently arising distributions; means and variances as useful summaries.
We may not expect our knowledge of the relevant probabilities to have the precision available for the examples in the previous chapter, but an approximate answer to the right question can be a reliable guide to good decisions. As statistician George Box said:‘All models are wrong, but some are useful.’
The next two chapters illustrate applications, loosely grouped under the chapter titles.
Brownian motion, and random walks
In 1827, the botanist Robert Brown observed that pollen particles suspended in liquid move around, apparently at random. Nearly eighty years later, Albert Einstein gave an explanation: the particles were constantly being buffeted by the molecules in the liquid. This movement is, of course, in three dimensions, but to build a satisfactory model, we first consider movement just along a straight line.
Suppose that each step is a jump of some fixed distance, sometimes left and sometimes right, independently each time. This notion is termed a random walk. The position after many jumps depends only on the difference between the numbers of jumps in each direction; the mean and variance of the distance from the start point are proportional to the number of jumps made.
Make a delicate computation: over a fixed time period, increase the frequency of the jumps, and decrease the distance jumped. With the correct balance between these two factors, the limit becomes continuous motion, the random distance moved having(by the Central Limit Theorem) a Gaussian distribution whose mean and variance are both proportional to the length of the time period. If movements left or right are equally likely, the mean will be zero.