Mathematical Modeling in the Social and Life Sciences

CHAPTER 10

Elementary Probability

The most important questions of life are, for the most part, really only questions of probability. Strictly speaking, one may even say that nearly all our knowledge is problematical; and in the small number of things which we are able to know with certainty, even in the mathematical sciences themselves, induction and analogy, the principal means of discovering truth, are based on probabilities, so that the entire system of human knowledge is connected with this theory.

—Pierre-Simon de Laplace

I. The Need for Probability Models

The deterministic and axiomatic models developed in earlier chapters show that both types of models can serve to give concise and precise descriptions of some real-world situations. Deterministic models have an added feature of being predictive in nature, while the best that axiomatic models seem to do is guarantee the existence or uniqueness of certain kinds of sets or functions.

The usefulness of a model increases if that model gives some new information not yet observed about the situation it is supposed to represent. The predictions of the model can be tested against what actually happens in the real world. Refinements can then be made in the model and better understanding gained of the real-world problem.

The deterministic models of Chapters 1–5 are typical of the type one sees in the natural, physical, and social sciences. They consist of systems of differential equations, the mathematical tool that has been most useful ...