Randomness

“ The desire to penetrate into the future and the ratios of some remarkable events, to the predictions of astrologers, of diviners and soothsayers, to presentiments and dreams, to the numbers and the days reputed lucky or unlucky, have given birth to a multitude of prejudices still very widespread. One does not reglect upon the great number of non-coincidences which have made no impression or which are unknown. However, it is necessary to be acquainted with them in order to appreciate the probability of the causes to which the coincidences are attributed. ”

“ Instead of the folly of the gods, there now is a randomizer with a scientific garb. If objects, and the molecules in our brains, have random aspects, then this would be neither determinism nor volition. ”

“ All that remains behind is an indistinct outline which can never be filled up. Under certain conditions rare occurrences are quite as probable as common ones. The main condition of the probability of such an event is that the rare occurrence should, from its nature and from the circumstances under which it occurs, be capable of being observed, and that the evidence of it should be recorded in the manner which I have already described. ”

“ In like manner, as the uncertainty of causes is discover'd by experience, which presents us with a view of contrary events, 'tis plain, that when we transfer the past to the future, the known to the unknown, every past experiment has the same weight, and that 'tis only a superior number of them, which can throw the ballance on any side. The possibility, therefore, which enters into every reasoning of this kind, is compos'd of parts, which are of the same nature both among themselves, and with those, that compose the opposite probability. ”

“ But we have no reason to suppose that matter, space, and time are necessarily parts of reality; of the ultimate Idea. Probability points rather to their being the pencil and paper with which we sketch it. ”

“ We want a replica of reality; this goes without saying. But wherever the reality is unattractive, or characterized, to be specific, by a bewildering fullness of lines, then it is that we need probability—or better still, verisimilitude. ”

“ If we have observed that two events have happened near together in time, we accept the occurrence of one as a sign that the other will follow. When we hear the factory whistle blow, we conclude that in a few minutes the workmen will pass our window on their way home. Such a conclusion is based upon a belief established by an inductive process. The degree of probability that it gives depends upon the number of times that it has been observed to act without failure. If we have seen two boys frequently together, the presence of one is a sign of the probable presence of the other. ”

“ These agreeing images unite together, and render the idea more strong and lively, not only than a mere fiction of the imagination, but also than any idea, which is supported by a lesser number of experiments. Each new experiment is as a new stroke of the pencil, which bestows an additional vivacity on the colours, without either multiplying or enlarging the figure. This operation of the mind has been so fully explain'd in treating of the probability of chance, that I need not here endeavour to render it more intelligible. Every past experiment may be consider'd as a kind of chance ”

“ An event occurring by chance may be better described as a coincidence from which we have no ground to infer a uniformity—the occurrence of a phenomenon in certain circumstances, without our having reason on that account to infer that it will happen again in those circumstances. ”

“ This connexion or constant conjunction sufficiently proves the one part to be the cause of the other. As the belief, which we have of any event, encreases or diminishes according to the number of chances or past experiments, 'tis to be consider'd as a compounded effect, of which each part arises from a proportionable number of chances or experiments.
Let us now join these three observations, and see what conclusion we can draw from them. To every probability there is an opposite possibility. ”

“ It is by such appeals to facts that all analogous superstitions, even in our day, attempt to justify themselves; the supposed evidence of experience is necessary to their hold on the mind. I quite admit that the influence of such coincidences would not be what it is, if strength were not lent to it by an antecedent presumption; but this is not peculiar to such cases; preconceived notions of probability form part of the explanation of many other cases of belief on insufficient evidence. ”

“ His invention connects the utmost stretch of possibility with the most plausible degree of probability, in a manner that equally surprises our fancy, persuades our judgment, and affects our heart. His composition always hastens to the most necessary point as its centre, and from that disseminates, to that leads back as rays, all secondary ones. Group, form, and contrast are subordinate to the event, and common-place ever excluded. ”

“ Moreover, the notion seems to prevail that these coincidences should occur with equal frequency to all persons; while, in fact, the law of probability provides for the most various distribution among individuals. However, the attempt, and it may be the sincere attempt, to apply proper conceptions of probability and improbability to such problems often fails, because of an unfortunate mental attitude which presents, with an outward acquiescence in the objective view of the problem, an inward conviction in which the subjective interpretation is really dominant ”

“ What we are to understand by a chance distribution is one in accordance with law, and one the nature of which can, for all practical purposes, be closely predicted." If the universe be pregnant with purpose, as we all wish to believe, why should not this purpose work itself out by an evolution process under law?—and if under law, why not the law of Probabilities? ”

“ We have thus far been concerned with Probability, determined by methods that take cognizance of Variations, and yield exact results, thereby affording a solid foundation for action. But the stage on which human action takes place is a superstructure into which emotion enters, we are guided on it less by Certainties and by Probabilities than by Assurance to a greater or lesser degree. ”

“ What we have found once to follow from any object, we conclude will for ever follow from it; and if this maxim be not always built upon as certain, ‘tis not for want of a sufficient number of experiments, but because we frequently meet with instances to the contrary; which leads us to the second species of probability, where there is a contrariety in our experience and observation. ”

“ Analogy is based upon the probability, that similar things have causes of the same kind and produce the same effects. This probability increase as the similitude becomes more perfect. Thus we judge without doubt that beings provided with the same organs, doing the same things, experience the same sensations, and are moved by the same desires. The probability that the animals which resemble us have sensations analogous to ours, although a little inferior to that which is relative to individuals of our species, is still exceedingly great ”

“ Coincidences, in general, are great stumbling blocks in the way of that class of thinkers who have been educated to know nothing of the theory of probabilities—that theory to which the most glorious objects of human research are indebted for the most glorious of illustration. ”

“ The paradoxes of what is called chance, or hazard, might themselves make a small volume. All the world understands that there is a long run, a general average; but great part of the world is surprised that this general average should be computed and predicted. There are many remarkable cases of verification; and one of them relates to the quadrature of the circle. I give some account of this and another. ”

“ These fit the men or women intended, but they also fit a crowd of other people. The description given by the scryer then may come right by a fortuitous coincidence, or may be too credulously recognised.
The complex of coincidences, however, could not be attributed to chance selection out of the whole possible field of conjecture. We must remember, too, that a series of such hits increases, at an enormous rate, the odds against accidental conjecture. ”

“ On the other hand, in experiences which happen extremely frequently and in the most diverse circumstances, and in which we always find the constant and unexceptional combination, the probability is very strong that we shall find the combination in future experiences also, and the probability of the conclusion approaches practical certainty. ”

“ As demonstration is subject to the controul of probability, so is probability liable to a new correction by a reflex act of the mind, wherein the nature of our understanding, and our reasoning from the first probability become our objects. ”

“ And if, now, suddenly the contents of that field are supposed to be understood and understandable only statistically 422and under the aspect of Probability—instead of under that of the a priori exactitude which the Baroque thinkers unanimously demanded—what does it mean? It means that the object of understanding is ourselves. ”

“ STUDY of the movements of events reveals dynamical, necessary sequences, and contemplation of the laws of probability, treated mathematically, generally involves a mental attitude at variance with theories of luck and premonition. ”

“ When a simple event or one composed of several simple events, as, for instance, in a game, has been repeated a great number of times the possibilities of the simple events which render most probable that which has been observed are those that observation indicates with the greatest probability; in the measure that the observed event is repeated this probability increases and would end by amounting to certainty if the numbers of repetitions should become infinite. ”

“ 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. ”

“ McGill grinned. "Don't be superstitious. And especially don't be anthropomorphic."
"Well, if it's the opposite of random, it's got to be a form of life."
"On what basis? All we know for certain is that random motions are being rearranged. A crystal, for example, is not life, but it's a non-random arrangement of particles.... ”

“ Since the usual correlations and connections become part of our unreflective expectations, and even seem, except to the psychologist, to form part of our data, it comes to be thought, mistakenly, that in such cases the data are unreal, whereas they are merely the causes of false inferences. The fact that correlations and connections of unusual kinds occur adds to the difficulty of inferring things from sense and of expressing physics in terms of sense-data. ”

“ In a word, instead of taking known events and deducing from them what has occurred, we must build up a fanciful explanation if it will only be consistent with known events. We can then test this explanation by any fresh facts which may arise. If they all fit into their places, the probability is that we are upon the right track, and with each fresh fact this probability increases in a geometrical progression until the evidence becomes final and convincing. ”

“ To all which we may add, that there was frequently a necessity to be very circumstantial and minute, in order to preserve and maintain that air of probability, which is necessary to be maintained in a story designed to represent real life ”

“ Every great law that is added to our store adds also to our conviction that the universe is run through with Mind. Even so-called Chance, which used to be the "bogie" behind Natural Selection, has now been found to illustrate—in the law of Probabilities—the absence of Chance. ”

“ If the number of experiments be very large, we may have precise information as to the value of the mean, but if our sample be small, we have two sources of uncertainty:— (1) owing to the “error of random sampling” the mean of our series of experiments deviates more or less widely from the mean of the population, and (2) the sample is not sufficiently large to determine what is the law of distribution of individuals. ”

“ We may arrive at them only by multiplying the observations or the experiences, so that the strange effects finally destroy reciprocally each other, the mean results putting in evidence those phenomena and their divers elements. The more numerous the number of observations and the less they vary among themselves the more their results approach the truth. We fulfil this last condition by the choice of the methods of observations, by the precision of the instruments, and by the care which we take to observe closely ”

“ If instead of drawing one bean we draw a handful at random and conclude that about of the handful are probably white, the reasoning is of the same sort. If, however, not knowing what proportion of white beans there are in the bag, we draw a handful at random and, finding 23 of the beans in the handful white, conclude that about 23 of those in the bag are white, we are rowing up the current of deductive sequence, and are concluding a rule from the observation of a result in a certain case. ”