ART. X.-Théorie Analytique des Probabilités. Par M. le Marquis de Laplace, &c. &c. 3ième édition. Paris. 1820.

N continuing our remarks upon the work of which the title

have not room to enter at length upon the subject. We have already discussed considerations of a practical character, tending to shew that upon several questions, in which recourse is actually had to the theory of probabilities, insufficiency of information produces effects prejudicial to the pecuniary interests of those concerned. This is indeed a strong point: we might urge any plan of prospective utility upon the English public, till we were tired, and without awakening the least attention. Nor would there be any reason to complain of such a result; for the present is an age of suggestions, and every person who can read and write has some scheme in hand, by which the community is to be advantaged: no wonder, then, that so few of the speculations in question have more than one investigator. But when we speak of the theory of probabilities, we bring forward a something upon which, right or wrong, many tens of millions of pounds sterling depend. The insurance offices, the friendly societies, all annuitants and all who hold life interests of any species-again, all who insure their goods from fire, or their ships from wreck-are visibly and immediately interested in the dissemination of correct principles upon probability in general. So much for that which actually is invested: now with regard to that which might be, let it be remembered, that whenever money is hazarded in commerce or manufactures, by those who would resign the possibility of more than average profit, if they might thereby be secured from the risk of disastrous loss, the desired arrangement is rendered impossible, by the want of knowledge how to apply the theory of probabilities, combined with the defect of methodized information upon the contingencies in question.

The name of the theory of probabilities is odious in the eyes of many, for, as all the world knows, it is the new phrase for the computation of chances, the instrument of gamblers, and, for a long time, of gamblers only; meaning, by that word, not the people who play with stocks and markets, but with cards, dice, and horses. Such an impression was the inevitable consequence of the course pursued by the earlier writers on the subject, who filled their books entirely with problems relative to games of chance. This was not so much a consequence of the nature of the subject, as of the state of mathematical knowledge at the

time games of chance, involving a given and comparatively small number of cases, are of easy calculation, and require only the application of simple methods; while questions of natural philosophy, or concerning the common affairs of life, involve very large numbers of cases, and require a more powerful analysis. Consequently, the older works abound with questions upon games of chance, while later writings begin to display the power of applying the very same principles to wider as well as more useful inquiries.

This objection to the tendency of the theory of probability, or the doctrine of chances, is as old as the time of De Moivre ; who was not, however, able to meet it, by extending the subject matter of his celebrated treatise. In the second edition, published in 1738, he writes thus, in his dedication to a Lord Carpenter: "There are many people in the world who are prepossessed with an opinion, that the doctrine of chances has a tendency to promote play; but they soon will be undeceived, if they think fit to look into the general design of this book. In the mean while, it will not be improper to inform them, that your lordship is pleased to espouse the patronage of this second edition," &c. &c. The general design of De Moivre's work appears to be, the analysis of every game of chance which prevailed in his time; and the author seems to have imagined that he could not attract attention to any other species of problems.

In reviewing the general design of the work of Laplace, we desire to make the description of a book mark the present state of a science. In any other point of view, it would be superfluous to give an account of a standard treatise, which is actually in the hands of a larger number of persons than are able to read it.

In considering simple questions of chances, we place ourselves, at the outset, in hypothetical possession of a set of circumstances, and attribute to ourselves exact and rigorous knowledge. We assume that we positively know every case that can arrive, and also that we can estimate the relative probabilities of the several cases. This of itself has a tendency to mislead the beginner, because these known circumstances are generally expressed by means of some simple gambling hypothesis. A set of balls which have been drawn, 83 white and 4 black, places us in the same position with regard to our disposition to expect white or black for the future, as that in which we should stand if we had observed 83 successful and 4 unsuccessful speculations in a matter of business: it matters nothing as to the amount of the chances for the future, whether the observed event be called the drawing of a white ball, or the acquirement of a profit. Never

theless, the abstraction of the idea of probability from the circumstances under which it is presented, sometimes throws a difficulty in the way.

The science of probability has also this in common with others, that the problems which most naturally present themselves in practice are of an inverse character, as compared with those which an elementary and deductive course first enables the student to solve. If we know that out of 1000 infants born, 900 live a year, it is sufficiently easy to understand why we say that it is nine to one any specified individual of them will live a year. But seeing that we can only arrive at such knowledge by observation, and also that such observation must be limited, there arises this very obvious preliminary question-Having registered a certain thousand infants, and found that, of that thousand, nine hundred were alive at the end of a year, what presumption arises from thence that something like the same proportion would obtain if a second thousand were registered? For instance, would it be wise to lay an even bet that the results of the second trial would exhibit something between 850 and 950, in place of 900? Or, to generalize the form of the question, let us imagine a thousand balls to have been drawn from a lottery containing an infinite number; of which it is found that there are 721 white, 116 red, and 163 black. We may then ask, what degree of presumption ought to be considered as established-1. That the contents of the lottery are all white, red, and black, and of no other colour? 2. That the white and red balls are distributed throughout the whole mass, nearly in the proportion of 721 white to 116 red? This is a question which must present itself previously to the deduction of any inference upon the probable results of future drawings: but at the same time, it is not of the most direct and easy class, requiring, in fact, the previous discussion of many methods which are subsequent in the order of application.

It is common to assume that any considerable number of observations will give a result nearly coinciding with the average of the whole. The constructors of the Northampton and Carlisle tables (see the last Number, p. 344) did not think it necessary to ask whether 2,400 and 861 cases of mortality would of themselves furnish a near approximation to the law which actually prevails in England. It had been long admitted, or supposed, that a considerable number of deaths (no definite number being specified) would present a table of mortality, such as might be depended upon for pecuniary transactions. It is true that such is the case; but the proposition is one requiring that sort of examination and demonstration which Laplace has

given. We shall not stop to rebut any conclusion which might be drawn against the utility of the theory, from the circumstance of common sense having felt for and attained some of its most elaborate results: but we shall stop to remark, that in the case of a speculation, so very delicate, so very liable to be misunderstood, and, above all, accessible to so small a part of the educated world, it is a great advantage that there exist such landmarks, as propositions which, though distant results of theory, yet coincide with the notions of the world at large, and are supposed to have evidence of their own.

When we have learnt that the result of analysis agrees with general opinion, in admitting the safety of relying upon a comparatively small number of cases to determine a general average, we then become disposed to rely on the same analysis for correctly determining the probable limits of accidental Яuctuation.

The two-fold object of the theory is, then, firstly, to determine the mean, or average state of things; secondly, to ascertain what degree of fluctuation may be reasonably expected. Let it be remarked, that the common theory of chances applies itself almost entirely to the first-mentioned problem: when we say that we determine the probability of an event to be two-sevenths, we mean, that, taking every possible case in which the said event can happen, we shall find that it will happen twice out of seven times. Such is then the general average: but, supposing that we select 700 possible cases out of the whole, it does not therefore become probable, or more likely than not, that the event shall happen precisely 200 times, and fail precisely 500 times. All that becomes very likely is, that the number of arrivals shall be nearly 200, and of non-arrivals nearly 500; and it is one of the most important objects of the theory, to ascertain within what limits there is a given amount of probability that the departure from the general average shall be contained.

The question thus enunciated is of no small practical importance, and to the neglect of it we must attribute the supposed necessity for the large capitals with which many undertakings are commenced. (See last Number, p. 342.) Let us imagine an insurance office to be founded, and, for the sake of simplicity, let it take no life except at the age of 30. Let the materials for its management consist in the examination of a register of 1,000 lives, which have been found to drop in the manner pointed out, say by the Carlisle table. The premium which should be demanded is then easily ascertained; but its security depends upon two circumstances-1. That the 1,000 lives so recorded, shall represent the general mortality. 2. That the

amount of business obtained by the office, shall be so large as to render their actual experience another representation of the same general average. Neither of these conditions can be precisely attained; some small allowance must be made for both; and the question is, what amount of additional premium is necessary to cover the risk of fluctuation?-what number of insured lives will be sufficient to begin with?-or, supposing that all risks are to be taken, what is the smallest capital upon which a commencement can prudently be made, without any security for a large amount of business?

Perhaps we could not in fewer words convey an idea of the different states of the science in the times of De Moivre and Laplace, than by stating, that the former could have ascertained the requisite premium, and that the latter could have made the necessary additions for fluctuation, &c.

We now pass from matters of business, as to which we can only say what might be done,-to questions connected with the sciences of observation and experiment, in which we can appeal to what has been done. In every branch of inquiry which involves the actual use of our physical senses, the repetition of a process will always afford a series of discordances, varying in amount with the method used, the skill of the observer, and the nature of the observation. If the observed discordances present anything like uniformity of character, we are naturally led to conclude, that they are not, properly speaking, the results of errors of observation, but of some unknown law, by which the predicted or expected result is modified. If the discrepancy merely arise from error of observation, we must suppose that it will be sometimes of one kind and sometimes of another; sometimes producing a result larger than might have been expected, and sometimes smaller. Now, having noticed a set of observations which do not agree, it is one of the first objects of the theory to settle what presumption should exist that the variations are accidental (that is, totally unregulated by apparent or discoverable law), or that they follow a law which then becomes the object of investigation. The case taken by Laplace, as an illustration, will do for the same purpose here. It was suspected that, independently of local fluctuations, the barometer was always a little higher in the morning than in the afternoon. To settle this point, four hundred days were chosen, in which the barometer was remarkably steady, not varying four millimetres in any one day. This was done to avoid the large fluctuations, which would have rendered the changes in question, if such there were, imperceptible. It was found that the sum of the heights of the barometer at nine in the morning, exceeded the sum

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