Probability theory

Author: P. Whittle
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Publish Date: 2003-01-01
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Fragment: The application of probability theory to games of chance is an obvious one. However, there are applications in science and technology which are just as clean-cut. As examples, we can quote the genetic mechanism of Mendelian inheritance (Sections 5.5 and 9.2), the operation of a telephone exchange (Section 10.4), or the decay of radioactive molecules (Section 10.4). In all these cases one can make valuable progress with a simple model, although it is only fair to add that a deeper study will demand something more complicated. In general, the physical sciences provide a rich source of interesting and well-defined probability problems: see, for example, some of the models of statistical mechanics (Sections 6.3 and 10.9) and Brownian motion (Section 10.11). The problems generated by the technological sciences can be just as interesting and no less fundamental: see, for example, the discussion of information theory (Sections 16.1-16.3) and of routing in a telephone network (Section 10.8). In encountering the 'natural variability' of biological problems one runs into rather more diffuse situations, but this variability makes a probabilistic approach all the more imperative, and one can construct probabilistic models of, say, population growth (Sections 6.4 and 10.4) and epidemics (Section 10.7) which have proved useful. One encounters natural variability in the human form if one tries to construct social or economic models, but again such models, inevitably probabilistic, prove useful. See, for example, the discussion of the Pareto distribution in Section 10.5. One of the most recent and fascinating applications of probability theory is to the field of control or, more generally, to that of sequential decision-making (Section 16.4). One might, for example, be wishing to hold an aircraft on course, despite the fact that random forces of one sort or another tend to divert it, or one might wish to keep a factory in a state of over-all efficient production despite the fact that so many future variables (such as demand for the product) must be uncertain. In either case, one must make a sequence of decisions (regarding course adjustment or factory management) in such a way as to ensure efficient running over a period, or even optimal running, in some well-defined sense. Moreover, these decisions must be taken in the face of an uncertain future. It should also be said that probability theory has its own flavour and intrinsic structure, quite apart from applications, as may be apparent from Chapters 2, 3, 12, 13, and 14 in particular. Just as for mathematics in general, people argue about the extent to which the theory is self-generating, or dependent upon applications to suggest the right direction and concepts. Perhaps either extreme view is incorrect; the search for an inner pattern and the search for a physical pattern are both powerful research tools, neither of them to be neglected.
2. The Empirical Basis
Certain experiments are nonreproducible in that, when repeated under standard conditions, they produce variable results. The classic example is that of coin-tossing: the toss being the experiment, resulting in the observation of a head or a tail. To take something less artificial, one might be observing the response of a rat to a certain drug, observation on another rat constituting repetition of the experiment. However uniform in constitution the experimental animals may be, one will certainly observe a variable response. The same variability would be found in, for example, lifetimes of electric lamps, crop yields, collisions of physical particles, or the number of telephone calls made over a given line on a given day of the week. This variability cannot always be dismissed as 'experimental error', which could presumably be explained and reduced, but may be something more fundamental. For instance, the next ejection of an electron from a hot metal filament is a definite event, whose time is not predictable on any physical theory yet developed. Probability theory can be regarded as an attempt to provide a quantitative basis for the discussion of such situations, or at least for some of them. One might despair of constructing a theory for phenomena whose essential quality is that of imprecision, but there is an empirical observation which gives the needed starting point. Suppose one tosses a coin repeatedly, keeping a record of the number of heads (n) in the first n tosses (n = 1, 2, 3, ...). Consider now the proportion of heads after n tosses: It is an empirical fact that p(n) varies with n much as in Fig. 1.1, which is derived from a genuine coin-tossing experiment. The values of p(n) show Figure 1.1. A graph of the proportions of heads thrown, p(n), in a sequence of n throws, from an actual coin-tossing experiment. Note the logarithmic scale for n. The figures are taken from Kerrich (1946), by courtesy of Professor Kerrich and his publishers.