The Art of Probability: For Scientists and Engineers

Book Preface

Every field of knowledge has its subject matter and its methods, along with a style for handling them. The field of Probability has a great deal of the Art component in it-not only is the subject matter rather different from that of other fields, but at present the techniques are not well organized into systematic methods. As a result each problem has to be “looked at in the right way” to make it easy to solve. Thus in probability theory there is a great deal of art in setting up the model, in solving the problem, and in applying the results back to the real world actions that will follow. It is necessary to include some of this art in any textbook that tries to prepare the reader to use probability in the real world of science and engineering rather than merely admire it as an abstract discipline and a branch of mathematics. It is widely agreed that art is best taught through concrete examples. Especially in teaching probability it is necessary to work many problems. Since the answers are already known the purpose of the Examples and Exercises cannot be to “get the answer” but to illustrate the methods and style of thinking. Hence the Examples in the text should be studied for the methods and style as well as for the results; also they often have educational value. Thus in solving the Exercises style should be considered as part of their purpose. It is not enough to merely give solutions to problems in probability. If the art is to be communicated to the reader then the initial approach-which is so vital in this field-must be carefully discussed. From where, for example, do the initial probabilities come? Only in this way can the reader learn this art-and most mathematically oriented text book simply ignore the source of the probabilities!

What is probability? I asked myself this question many years ago, and found that various authors gave different answers. I found that there were several main schools of thought with many variations. First, there were the frequentists who believe that probability is the limiting ratio of the successes divided by the total number of trials (each time repeating essentially the same situation). Since I have a scientific-engineering background, this approach, when examined in detail, seemed to me to be non-operational and furthermore excluded important and interesting situations.

Second, there are those who think that there is a probability to be attached to a single, unique event without regard to repetitions. Via the law of large numbers they deduce the frequency approach as something that is likely, but not sure.

Third, I found, for example, that the highly respected probabilist di Finetti [dF, p. x] wrote at the opening of his two volume treatise,

Fourth, I found that mathematicians tend to simply postulate a Borel family of sets with suitable properties (often called a u-algebra or a Borel field) for the sample space of events and assign a measure over the field which is the corresponding probability. But the main problems in using probability theory are the choice of the sample space of events and the assigment of probability to the events! Some highly respected authors like Feller [F, p. x] and Kac [K, p. 24] were opposed to this measure theoretic mathematical approach to probability; both loudly proclaimed that probability is not a branch of measure theory, yet both in their turn seemed to me to adopt a formal mathematical approach, as if probability were merely a branch of mathematics and not an independent field.

Fifth, I also found that there is a large assortment of personal probabilists, the most prominent being the Bayesians, at least in volume of noise. Just what the various kinds of Bayesians are proclaiming is not always clear to me, and at least one said that while nothing new and testable is produced still it is “the proper way to think about probability”.

Finally, there were some authors who were very subjective about probability, seeming to say that each person had their own probabilities and there need be little relationship between their beliefs; possibly true in some situations but hardly scientific.