[7] The best example I can recall of the distinction between judging from the subjective and the objective side, in such cases as these, occurred once in a railway train. I met a timid old lady who was in much fear of accidents. I endeavoured to soothe her on the usual statistical ground of the extreme rarity of such events. She listened patiently, and then replied, “Yes, Sir, that is all very well; but I don't see how the real danger will be a bit the less because I don't believe in it.”

[8] This would still hold of empirical laws which may be capable of being broken: we now have very much shifted the word, to denote an ultimate law which it is supposed cannot be broken.

CHAPTER VII.

THE RULES OF INFERENCE IN PROBABILITY.

§ 1. In the previous chapter, an investigation was made into what may be called, from the analogy of Logic, Immediate Inferences. Given that nine men out of ten, of any assigned age, live to forty, what could be inferred about the prospect of life of any particular man? It was shown that, although this step was very far from being so simple as it is frequently supposed to be, and as the corresponding step really is in Logic, there was nevertheless an intelligible sense in which we might speak of the amount of our belief in any one of these ‘proportional propositions,’ as they may succinctly be termed, and justify that amount. We must now proceed to the consideration of inferences more properly so called, I mean inferences of the kind analogous to those which form the staple of ordinary logical treatises. In other words, having ascertained in what manner particular propositions could be inferred from the general propositions which included them, we must now examine in what cases one general proposition can be inferred from another. By a general proposition here is meant, of course, a general proposition of the statistical kind contemplated in Probability. The rules of such inference being very few and simple, their consideration will not detain us long. From the data now in our possession we are able to deduce the rules of probability given in ordinary treatises upon the science. It would be more correct to say that we are able to deduce some of these rules, for, as will appear on examination, they are of two very different kinds, resting on entirely distinct grounds. They might be divided into those which are formal, and those which are more or less experimental. This may be otherwise expressed by saying that, from the kind of series described in the first chapters, some rules will follow necessarily by the mere application of arithmetic; whilst others either depend upon peculiar hypotheses, or demand for their establishment continually renewed appeals to experience, and extension by the aid of the various resources of Induction. We shall confine our attention at present principally to the former class; the latter can only be fully understood when we have considered the connection of our science with Induction.

§ 2. The fundamental rules of Probability strictly so called, that is the formal rules, may be divided into two classes,—those obtained by addition or subtraction on the one hand, corresponding to what are generally termed the connection of exclusive or incompatible events;[1] and those obtained by multiplication or division, on the other hand, corresponding to what are commonly termed dependent events. We will examine these in order.

(1) We can make inferences by simple addition. If, for instance, there are two distinct properties observable in various members of the series, which properties do not occur in the same individual; it is plain that in any batch the number that are of one kind or the other will be equal to the sum of those of the two kinds separately. Thus 36.4 infants in 100 live to over sixty, 35.4 in 100 die before they are ten;[2] take a large number, say 10,000, then there will be about 3640 who live to over sixty, and about 3540 who do not reach ten; hence the total number who do not die within the assigned limits will be about 2820 altogether. Of course if these proportions were accurately assigned, the resultant sum would be equally accurate: but, as the reader knows, in Probability this proportion is merely the limit towards which the numbers tend in the long run, not the precise result assigned in any particular case. Hence we can only venture to say that this is the limit towards which we tend as the numbers become greater and greater.

This rule, in its general algebraic form, would be expressed in the language of Probability as follows:—If the chances of two exclusive or incompatible events be respectively ¹/_m and ¹/_n the chance of one or other of them happening will be ¹/_m + ¹/_n or ^m + n/_mn. Similarly if there were more than two events of the kind in question. On the principles adopted in this work, the rule, when thus algebraically expressed, means precisely the same thing as when it is expressed in the statistical form. It was shown at the conclusion of the last chapter that to say, for example, that the chance of a given event happening in a certain way is ¹/₆, is only another way of saying that in the long run it does tend to happen in that way once in six times.

It is plain that a sort of corollary to this rule might be obtained, in precisely the same way, by subtraction instead of addition. Stated generally it would be as follows:—If the chance of one or other of two incompatible events be ¹/_m and the chance of one alone be ¹/_n, the chance of the remaining one will be ¹/_m − ¹/_n or ^n − m/_nm.

For example, if the chance of any one dying in a year is ¹/₁₀, and his chance of dying of some particular disease is ¹/₁₀₀, his chance of dying of any other disease is ⁹/₁₀₀.