Probability

“ In short, to express the proper state of our belief, not one number but two are requisite, the first depending on the inferred probability, the second on the amount of knowledge on which that probability is based. [1] It is true that when our knowledge is very precise, when we have made many drawings from the bag, or, as in most of the examples in the books, when the total contents of the bag are absolutely known, the number which expresses the uncertainty of the assumed probability and its liability to be changed by further experience may become insignificant, or utterly vanish. ”

“ The probability of events, as calculated from their mere frequency in past experience, affords a less secure basis for practical guidance than their probability as deduced from an equally accurate knowledge of the frequency of occurrence of their causes.
The generalization that an event occurs in ten out of every hundred cases of a given description, is as real an induction as if the generalization were that it occurs in all cases. ”

“ But it has probabilities of different amounts in regard to different premisses. When we speak, elliptically, of the probability of a proposition, we mean its probability in relation to all our relevant knowledge. A proposition in probability cannot be refuted by mere observation: improbable things may happen and probable things may fail to happen. Nor is an estimate of probability relevant to given evidence proved wrong when further evidence alters the probability. ”

“ The common theory, therefore, of the calculation of chances, appears to be tenable. Even when we know nothing except the number of the possible and mutually excluding contingencies, and are entirely ignorant of their comparative frequency, we may have grounds, and grounds numerically appreciable, for acting on one supposition rather than on another; and this is the meaning of Probability. ”

“ No inductive conclusions are more than probable, and I adopt the opinion that the theory of probability is an essential part of logical method, so that the logical value of every inductive result must be determined consciously or unconsciously, according to the principles of the inverse method of probability.
The phenomena of nature are commonly manifested in quantities of time, space, force, energy, &c., and the observation, measurement, and analysis of the various quantitative conditions or results involved, even in a simple experiment, demand much employment of systematic procedure. ”

“ Such a law of average recurrence is the basis of Probability. Chance being the fact of coincidence without assignable cause, Probability is expectation based on the average frequency of its happening.
§ 2. Probability is an ambiguous term. Usually, when we say that an event is 'probable,' we mean that it is more likely than not to happen. But, scientifically, an event is probable if our expectation of its occurrence is less than certainty, as long as the event is not impossible. Probability, thus conceived, is represented by a fraction. ”

“ If the fact has no probability itself, such as the drawing of a number from an urn which contains an infinity of them, that which it acquires by the testimonies decreases according to the continued product of the veracity of the witnesses. ”

“ Amount of Probability.—The analogy and diversity, when they come thus into competition and the arguments from the one conflict with those of the other, must be weighed against each other. The extent of the resemblance, compared with the extent of the difference, gives the amount of probability on one side or the other, so far as these elements are known. If any region lies unexplored, we can infer nothing with certainty or probability as to that. ”

“ According to what has been said, the idea of probability essentially belongs to a kind of inference which is repeated indefinitely. An individual inference must be either true or false, and can show no effect of probability; and, therefore, in reference to a single case considered in itself, probability can have no meaning. ”

“ But to us, probability is the very guide of life. ”

“ The application of the principle which we have just expounded to the various questions of probability requires methods whose investigation has given birth to several methods of analysis and especially to the theory of combinations and to the calculus of finite differences. ”

“ According to the view here taken, these are the only fundamental rules for the calculation of chances. An additional one, derived from a different conception of probability, is given in some treatises, which if it be sound might be made the basis of a theory of reasoning. Being, as I believe it is, absolutely absurd, the consideration of it serves to bring us to the true theory; and it is for the sake of this discussion, which must be postponed to the next number, that I have brought the doctrine of chances to the reader's attention at this early stage of our studies of the logic of science. ”

“ Chapter IX. PROBABLE INFERENCE TO PARTICULARS—THE MEASUREMENT OF PROBABILITY.
Undoubtedly there are degrees of probability. Not only do we expect some events with more confidence than others: we may do so, and our confidence may be misplaced: but we have reason to expect some with more confidence than others. There are different degrees of rational expectation. Can those degrees be measured numerically?
The question has come into Logic from the mathematicians. The calculation of Probabilities is a branch of Mathematics. ”

“ Hence, though not a necessary, yet a most usual condition for calculating the probability of a fact is, that we should possess a specific knowledge of the proportion which the cases in which facts of the particular sort occur bear to the cases in which they do not occur. ”

“ An important remark which pertains to great generality of analysis, and which permits us to extend this method to formulae and to equations of difference which the theory of probability presents most frequently, is that the series to which one comes by supposing the limits of the definite integrals to be real and positive take place equally in the case where the equation which determines these limits has only negative or imaginary roots. ”

“ Those philosophers, who have divided human reason into knowledge and probability, and have defin'd the first to be that evidence, which arises from the comparison of ideas, are oblig'd to comprehend all our arguments from causes or effects under the general term of probability. ”

“ I present here without the aid of analysis the principles and general results of this theory, applying them to the most important questions of life, which are indeed for the most part only problems of probability. Strictly speaking it may even be said 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, the principal means for ascertaining truth—induction and analogy—are based on probabilities ”

“ The application of the calculus of probabilities to natural philosophy consists, then, in determining analytically the probability of the values of these functions and in choosing their indeterminant coefficients in such a manner that the law of this probability should be most rapidly descending. ”

“ Hence our surprise, though, as stated above, having no proper claim to admission into the science of Probability, is such a constant and regular accompaniment of that which Probability is concerned with, that notice must often be taken of it. ”

“ The belief, which attends the probability, is a compounded effect, and is form'd by the concurrence of the several effects, which proceed from each part of the probability. Since therefore each part of the probability contributes to the production of the belief, each part of the possibility must have the same influence on the opposite side ”

“ If an event may coincide with two or more other independent events, the probability that they will together be a sign of it, is found by multiplying together the fractions representing the improbability that each is a sign of it, and subtracting the product from unity. ”

“ Determining these ratios and comparing them with a great number of observations if one finds that they constantly satisfy it, the probability of the causes may increase to the point of equalling that of facts in regard to which there is no doubt. ”