Pierre Simon Laplace Philosophical Essay On Probabilities Of Compound

Laplace's Essay on Probabilities

A PHILOSOPHICAL ESSAY ON PROBABILITIES.

BY PIERRE SIMON, MARQUIS DE LAPLACE. TRANSLATED FROM THE SIXTH FRENCH EDITION BY FREDERICK WILSON TRUSCOTT, PH.D. (HARV.), Professor of Germanic Languages in the West Virginia. University, FREDERICK LINCOLN EMORY, M.E. (WOR. POLY. INST.), Professor of Mechanics and Applied Mathematics in the West Virginia University ; Mem. Amer. Soc. Mtch. Eng. NEW YORK: JOHN WILEY & SONS. LONDON : CHAPMAN & HALL, LIMITED. Copyright, 1902, by F. W. TRUSCOTT and F. L. EMORY. TABLE OF CONTENTS. PART I. A PHILOSOPHICAL ESSAY ON PROBABILITIES. CHAPTER I. Introduction 1CHAPTER II. Concerning Probability 3CHAPTER III. General Principles of the Calculus of Probabilities 11CHAPTER IV. Concerning Hope 20CHAPTER V. Analytical Methods of the Calculus of Probabilities 26 PART II. APPLICATION OF THE CALCULUS OF PROBABILITIES. CHAPTER VI. Games of Chance 53CHAPTER VII. Concerning the Unknown Inequalities which may Exist among Chances Supposed to be Equal 56CHAPTER VIII. Concerning the Laws of Probability which result from the Indefinite Multiplication of Events 6CHAPTER IX. Application of the Calculus of Probabilities to Natural Philosophy 73CHAPTER X. Application of the Calculus of Probabilities to the Moral Sciences 107CHAPTER XI Concerning the Probability of Testimonies 109CHAPTER XII. Concerning the Selections and Decisions of Assemblies 126CHAPTER XIII. Concerning the Probability of the Judgments of Tribunals 132CHAPTER XIV. Concerning Tables of Mortality, and the Mean Durations of Life, Marriage, and Some Associations 140CHAPTER XV. Concerning the Benefits of Institutions which Depend upon the Probability of Events 149CHAPTER XVI. Concerning Illusions in the Estimation of Probabilities 160CHAPTER XVII. Concerning the Various Means of Approaching Certainty 176CHAPTER XVIII. Historical Notice of the Calculus of Probabilities to 1816 185
CHAPTER I. INTRODUCTION. THIS philosophical essay is the development of a lecture on probabilities which I delivered in 1795 to the normal schools whither I had been called, by a decree of the national convention, as professor of mathematics with Lagrange. I have recently published upon the same subject a work entitled The Analytical Theory of Probabilities. 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; 2 A PHILOSOPHICAL ESSAY ON PROBABILITIES. so that the entire system of human knowledge is con- nected with the theory set forth in this essay. Doubt- less it will be seen here with interest that in considering, even in the eternal principles of reason, justice, and humanity, only the favorable chances which are con- stantly attached to them, there is a great advantage in following these principles and serious inconvenience in departing from them: their chances, like those favor- able to lotteries, always end by prevailing in the midst of the vacillations of hazard. I hope that the reflec- tions given in this essay may merit the attention of philosophers and direct it to a subject so worthy of engaging their minds. ToC3 CHAPTER II. CONCERNING PROBABILITY. ALL events, even those which on account of their insignificance do not seem to follow the great laws of nature, are a result of it just as necessarily as the revolu- tions of the sun. In ignorance of the ties which unite such events to the entire system of the universe, they have been made to depend upon final causes or upon hazard, according as they occur and are repeated with regularity, or appear without regard to order ; but these imaginary causes have gradually receded with the widening bounds of knowledge and disappear entirely before sound philosophy, which sees in them only the expression of our ignorance of the true causes. Present events are connected with preceding ones by a tie based upon the evident principle that a thing cannot occur without a cause which produces it. This axiom, known by the name of the principle of sufficient reason, extends even to actions which are considered indifferent ; the freest will is unable without a determi- native motive to give them birth; if we assume two positions with exactly similar circumstances and find that the will is active in the one and inactive in the 4 A PHILOSOPHICAL ESSAY ON PROBABILITIES. other, we say that its choice is an effect without a cause. It is then, says Leibnitz, the blind chance of the Epicureans. The contrary opinion is an illusion of the mind, which, losing sight of the evasive reasons of the choice of the will in indifferent things, believes that choice is determined of itself and without motives. We ought then to regard the present state of the universe as the effect of its anterior state and as the cause of the one which is to follow. Given for one instant an intelligence which could comprehend all the forces by which nature is animated and the respective situation of the beings who compose it an intelligence sufficiently vast to submit these data to analysis it would embrace in the same formula the movements of the greatest bodies of the universe and those of the lightest atom; for it, nothing would be uncertain and the future, as the past, would be present to its eyes. The human mind offers, in the perfection which it has been able to give to astronomy, a feeble idea of this in- telligence. Its discoveries in mechanics and geometry, added to that of universal gravity, have enabled it to comprehend in the same analytical expressions the past and future states of the system of the world. Applying the same method to some other objects of its knowledge, it has succeeded in referring to general laws observed phenomena and in foreseeing those which given circumstances ought to produce. All these efforts in the search for truth tend to lead it back continually to the vast intelligence which we have just mentioned, but from which it will always remain infinitely removed. This tendency, peculiar to the human race, is that which renders it superior to animals; and their progress CONCERNING PROBABILITY. 5 in this respect distinguishes nations and ages and con- stitutes their true glory. Let us recall that formerly, and at no remote epoch, an unusual rain or an extreme drought, a comet having in train a very long tail, the eclipses, the aurora borealis, and in general all the unusual phenomena were regarded as so many signs of celestial wrath. Heaven was invoked in order to avert their baneful influence. No one prayed to have the planets and the sun arrested in their courses: observation had soon made apparent the futility of such prayers. But as these phenomena, occurring and disappearing at long intervals, seemed to oppose the order of nature, it was supposed that Heaven, irritated by the crimes of the earth, had created them "to announce its vengeance." Thus the long tail of the comet of 1456 spread terror through Europe, already thrown into consternation by the rapid successes of the Turks, who had just over- thrown the Lower Empire. This star after four revolu- tions has excited among us a very different interest. The knowledge of the laws of the system of the world acquired in the interval had dissipated the fears begotten by the ignorance of the true relationship of man to the universe; and Halley, having recognized the identity of this comet with those of the years 1531, 1607, and 1682, announced its next return for the end of the year 1758 or the beginning of the year 1759. The learned world awaited with impatience this return which was to confirm one of the greatest discoveries that have been made in the sciences, and fulfil the prediction of Seneca when he said, in speaking of the revolutions of those stars which fall from an enormous 6 A PHILOSOPHICAL ESSAY ON PROBABILITIES. height: "The day will come when, by study pursued through several ages, the things now concealed will appear with evidence; and posterity will be astonished that truths so clear had escaped us." Clairaut then undertook to submit to analysis the perturbations which the comet had experienced by the action of the two great planets, Jupiter and Saturn; after immense cal- culations he fixed its next passage at the perihelion toward the beginning of April, 1759, which was actually verified by observation. The regularity which astronomy shows us in the movements of the comets doubtless exists also in all phenomena. The curve described by a simple molecule of air or vapor is regulated in a manner just as certain as the planetary orbits; the only difference between them is that which comes from our ignorance. Probability is relative, in part to this ignorance, in part to our knowledge. We know that of three or a greater number of events a single one ought to occur ; but nothing induces us to believe that one of them will occur rather than the others. In this state of indecision it is impossible for us to announce their occurrence with certainty. It is, however, probable that one of these events, chosen at will, will not occur because we see several cases equally possible which exclude its occur- rence, while only a single one favors it. The theory of chance consists in reducing all the events of the same kind to a certain number of cases equally possible, that is to say, to such as we may be equally undecided about in regard to their existence, and in determining the number of cases favorable to the event whose probability is sought. The ratio of CONCERNING PROBABILITY. 7 this number to that of all the cases possible is the measure of this probability, which is thus simply a fraction whose numerator is the number of favorable cases and whose denominator is the number of all the cases possible. The preceding notion of probability supposes that, in increasing in the same ratio the number of favorable cases and that of all the cases possible, the probability remains the same. In order to convince ourselves let us take two urns, A and B, the first containing four white and two black balls, and the second containing only two white balls and one black one. We may imagine the two black balls of the first urn attached by a thread which breaks at the moment when one of them is seized in order to be drawn out, and the four white balls thus forming two similar systems. All the chances which will favor the seizure of one of the balls of the black system will lead to a black ball. If we conceive now that the threads which unite the balls do not break at all, it is clear that the number of possible chances will not change any more than that of the chances favorable to the extraction of the black balls; but two balls will be drawn from the urn at the same time; the probability of drawing a black ball from the urn A will then be the same as at first. But then we have obviously the case of urn B with the single differ- ence that the three balls of this last urn would be replaced by three systems of two balls invariably con- nected. When all the cases are favorable to an event the probability changes to certainty and its expression becomes equal to unity. Upon this condition, certainty 8 A PHILOSOPHICAL ESSAY ON PROBABILITIES. and probability are comparable, although there may be an essential difference between the two states of the mind when a truth is rigorously demonstrated to it, or when it still perceives a small source of error. In things which are only probable the difference of the data, which each man has in regard to them, is one of the principal causes of the diversity of opinions which prevail in regard to the same objects. Let us suppose, for example, that we have three urns, A, B, C, one of which contains only black balls while the two others contain only white balls ; a ball is to be drawn from the urn C and the probability is demanded that this ball will be black. If we do not know which of the three urns contains black balls only, so that there is no reason to believe that it is C rather than B or A, these three hypotheses will appear equally possible, and since a black ball can be drawn only in the first hypothesis, the probability of drawing it is equal to one third. If it is known that the urn A contains white balls only, the indecision then extends only to the urns B and C, and the probability that the ball drawn from the urn C will be black is one half. Finally this probability changes to certainty if we are assured that the urns A and B contain white balls only. It is thus that an incident related to a numerous assembly finds various degrees of credence, according to the extent of knowledge of the auditors. If the man who reports it is fully convinced of it and if, by his position and character, he inspires great confidence, his statement, however extraordinary it may be, will have for the auditors who lack information the same degree of probability as an ordinary statement made CONCERNING PROBABILITY. 9 by the same man, and they will have entire faith in it. But if some one of them knows that the same incident is rejected by other equally trustworthy men, he will be in doubt and the incident will be discredited by the enlightened auditors, who will reject it whether it be in regard to facts well averred or the immutable laws of nature. It is to the influence of the opinion of those whom the multitude judges best informed and to whom it has been accustomed to give its confidence in regard to the most important matters of life that the propagation of those errors is due which in times of ignorance have covered the face of the earth. Magic and astrology offer us two great examples. These errors inculcated in infancy, adopted without examination, and having for a basis only universal credence, have maintained themselves during a very long time; but at last the progress of science has destroyed them in the minds of enlightened men, whose opinion consequently has caused them to disappear even among the common people, through the power of imitation and habit which had so generally spread them abroad. This power, the richest resource of the moral world, establishes and conserves in a whole nation ideas entirely contrary to those which it upholds elsewhere with the same authority. What indulgence ought we not then to have for opinions different from ours, when this differ- ence often depends only upon the various points of view where circumstances have placed us! Let us enlighten those whom we judge insufficiently instructed; but first let us examine critically our own opinions and weigh with impartiality their respective probabilities. 10 A PHILOSOPHICAL ESSAY ON PROBABILITIES. The difference of opinions depends, however, upon the manner in which the influence of known data is determined. The theory of probabilities holds to con- siderations so delicate that it is not surprising that with the same data two persons arrive at different results, especially in very complicated questions. Let us examine now the general principles of this theory. ToCCHAPTER III. THE GENERAL PRINCIPLES OF THE CALCULUS OF PROBABILITIES. First Principle. The first of these principles is the definition itself of probability, which, as has been seen, is the ratio of the number of favorable cases to that of all the cases possible. Second Principle. But that supposes the various cases equally possible. If they are not so, we will determine first their respective possibilities, whose exact appreciation is one of the most delicate points of the theory of chance. Then the probability will be the sum of the possibilities of each favorable case. Let us illustrate this principle by an example. Let us suppose that we throw into the air a large and very thin coin whose two large opposite faces, which we will call heads and tails, are perfectly similar. Let us find the probability of throwing heads at least one time in two throws. It is clear that four equally possible cases may arise, namely, heads at the first and at the second throw; heads at the first throw and tails at the second; tails at the first throw and heads at the second; finally, tails at both throws. The first 12 A PHILOSOPHICAL ESSAY ON PROBABILITIES. three cases are favorable to the event whose probability is sought; consequently this probability is equal to 3/4; so that it is a bet of three to one that heads will be thrown at least once in two throws. We can count at this game only three different cases, namely, heads at the first throw, which dispenses with throwing a second time; tails at the first throw and heads at the second; finally, tails at the first and at the second throw. This would reduce the probability to 2/3 if we should consider with d'Alembert these three cases as equally possible. But it is apparent that the probability of throwing heads at the first throw is 1/2, while that of the two other cases is J, the first case being a simple event which corresponds to two events combined: heads at the first and at the second throw, and heads at the first throw, tails at the second. If we then, conforming to the second principle, add the possibility f of heads at the first throw to the possi- bility J of tails at the first throw and heads at the second, we shall have f for the probability sought, which agrees with what is found in the supposition when we play the two throws. This 'supposition does not change at all the chance of that one who bets on this event; it simply serves to reduce the various cases to the cases equally possible. Third Principle. One of the most important points of the theory of probabilities and that which lends the most to illusions is the manner in which these prob- abilities increase or diminish by their mutual combina- tion. If the events are independent of one another, the probability of their combined existence is the product of their respective probabilities. Thus the probability CALCULUS OF PROBABILITIES. 13 of throwing one ace with a single die is ^; that of throwing two aces in throwing two dice at the same time is --$. Each face of the one being able to com- bine with the six faces of the other, there are in fact thirty-six equally possible cases, among which one single case gives two aces. Generally the probability that a simple event in the same circumstances will occur consecutively a given number of times is equal to the probability of this simple event raised to the power indicated by this number. Having thus the successive powers of a fraction less than unity diminishing without ceasing, an event which depends upon a series of very great probabilities may become extremely improbable. Suppose then an incident be transmitted to us by twenty witnesses in such manner that the first has transmitted it to the second, the second to the third, and so on. Suppose again that the probability of each testimony be equal to the fraction T 9 ; that of the incident resulting from the testimonies will be less than . We cannot better compare this diminution of the probability than with the extinction of the light of objects by the interposition of several pieces of glass. A relatively small number of pieces suffices to take away the view of an object that a single piece allows us to perceive in a distinct manner. The historians do not appear to have paid sufficient attention to this degradation of the probability of events when seen across a great number of successive generations; many historical events reputed as certain would be at least doubtful if they were submitted to this test. In the purely mathematical sciences the most distant consequences participate in the certainty of the princi- 14 A PHILOSOPHICAL ESSAY ON PROBABILITIES. pie from which they are derived. In the applications of analysis to physics the results have all the certainty of facts or experiences. But in the moral sciences, where each inference is deduced from that which pre- cedes it only in a probable manner, however probable these deductions may be, the chance of error increases with their number and ultimately surpasses the chance of truth in the consequences very remote from the principle. Fourth Principle. When two events depend upon each other, the probability of the compound event is the product of the probability of the first event and the probability that, this event having occurred, the second will occur. Thus in the preceding case of the three urns A, B, C, of which two contain only white balls and one contains only black balls, the probability of drawing a white ball from the urn C is f , since of the three urns only two contain balls of that color. But when a white ball has been drawn from the urn C, the indecision relative to that one of the urns which contain only black balls extends only to the urns A and B; the probability of drawing a white ball from the urn B is ; the product of \ by , or , is then the probability of drawing two white balls at one time from the urns B and C. We see by this example the influence of past events upon the probability of future events. For the prob- ability of drawing a white ball from the urn B, which primarily is f, becomes \ when a white ball has been drawn from the urn C ; it would change to certainty if a black ball had been drawn from the same urn. We will determine this influence by means of the follow- CALCULUS OF PROBABILITIES. 15 ing principle, which is a corollary of the preceding one. Fifth Principle. If we calculate a priori the prob- ability of the occurred event and the probability of an event composed of that one and a second one which is expected, the second probability divided by the first will be the probability of the event expected, drawn from the observed event. Here is presented the question raised by some philosophers touching the influence of the past upon the probability of the future. Let us suppose at the play of heads and tails that heads has occurred oftener than tails. By this alone we shall be led to believe that in the constitution of the coin there is a secret cause which favors it. Thus in the conduct of life constant happiness is a proof of competency which should induce us to employ preferably happy persons. But if by the unreliability of circumstances we are con- stantly brought back to a state of absolute indecision, if, for example, we change the coin at each throw at the play of heads and tails, the past can shed no light upon the future and it would be absurd to take account of it. Sixth Principle. Each of the causes to which an observed event may be attributed is indicated with just as much likelihood as there is probability that the event will take place, supposing the event to be constant. The probability of the existence of any one of these causes is then a fraction whose numerator is the prob- ability of the event resulting from this cause and whose denominator is the sum of the similar probabilities relative to all the causes; if these various causes, con- sidered a priori, are unequally probable, it is necessary, 16 A PHILOSOPHICAL ESSAY ON PROBABILITIES. in place of the probability of the event resulting from each cause, to employ the product of this probability by the possibility of the cause itself. This is the funda- mental principle of this branch of the analysis of chances which consists in passing from events to causes. This principle gives the reason why we attribute regular events to a particular cause. Some philosophers have thought that these events are less possible than others and that at the play of heads and tails, for example, the combination in which heads occurs twenty successive times is less easy in its nature than those where heads and tails are mixed in an irregular manner. But this opinion supposes that past events have an influence on the possibility of future events, which is not at all admissible. The regular combinations occur more rarely only because they are less numerous. If we seek a cause wherever we perceive symmetry, it is not that we regard a symmetrical event as less possible than the others, but, since this event ought to be the effect of a regular cause or that of chance, the first of these suppositions is more probable than the second. On a table we see letters arranged in this order, Constantinople, and we judge that this arrange- ment is not the result of chance, not because it is less possible than the others, for if this word were not employed in any language we should not suspect it came from any particular cause, but this word being in use among us, it is incomparably more probable that some person has thus arranged the aforesaid letters than that this arrangement is due to chance. This is the place to define the word extraordinary. We arrange in our thought all possible events in various CALCULUS OF PROBABILITIES. 17 classes ; and we regard as extraordinary those classes which include a very small number. Thus at the play of heads and tails the occurrence of heads a hundred successive times appears to us extraordinary because of the almost infinite number of combinations which may occur in a hundred throws; and if we divide the com- binations into regular series containing an order easy to comprehend, and into irregular series, the latter are incomparably more numerous. The drawing of a white ball from an urn which among a million balls contains only one of this color, the others being black, would appear to us likewise extraordinary, because we form only two classes of events relative to the two colors. But the drawing of the number 475813, for example, from an urn that contains a million numbers seems to us an ordinary event; because, comparing individually the numbers with one another without dividing them into classes, we have no reason to believe that one of them will appear sooner than the others. From what precedes, we ought generally to conclude that the more extraordinary the event, the greater the need of its being supported by strong proofs. For those who attest it, being able to deceive or to have been deceived, these two causes are as much more probable as the reality of the event is less. We shall see this particularly when we come to speak of the probability of testimony. Seventh Principle. The probability of a future event is the sum of the products of the probability of each cause, drawn from the event observed, by the prob- ability that, this cause existing, the future event will 18 A PHILOSOPHICAL ESSAY ON PROBABILITIES. occur. The following example will illustrate this principle. Let us imagine an urn which contains only two balls, each of which may be either white or black. One of these balls is drawn and is put back into the urn before proceeding to a new draw. Suppose that in the first two draws white balls have been drawn; the prob- ability of again drawing a white ball at the third draw is required. Only two hypotheses can be made here : either one of the balls is white and the other black, or both are white. In the first hypothesis the probability of the event observed is J; it is unity or certainty in the second. Thus in regarding these hypotheses as so many causes, we shall have for the sixth principle % and | for their respective probabilities. But if the first hypothesis occurs, the probability of drawing a white ball at the third draw is ^ ; it is equal to certainty in the second hypothesis ; multiplying then the last probabilities by those of the corresponding hypotheses, the sum of the products, or T 9 ^, will be the probability of drawing a white ball at the third draw. When the probability of a single event is unknown we may suppose it equal to any value from zero to unity. The probability of each of these hypotheses, drawn from the event observed, is, by the sixth prin- ciple, a fraction whose numerator is the probability of the event in this hypothesis and whose denominator is the sum of the similar probabilities relative to all the hypotheses. Thus the probability that the possibility of the event is comprised within given limits is the sum of the fractions comprised within these limits. Now if CALCULUS OF PROBABILITIES. 19 we multiply each fraction by the probability of the future event, determined in the corresponding hypothe- sis, the sum of the products relative to all the hypotheses will be, by the seventh principle, the probability of the future event drawn from the event observed. Thus we find that an event having occurred successively any number of times, the probability that it will happen again the next time is equal to this number increased by unity divided by the same number, increased by two units. Placing the most ancient epoch of history at five thousand years ago, or at 182623 days, and the sun having risen constantly in the interval at each revolution of twenty-four hours, it is a bet of 1826214 to one that it will rise again to-morrow. But this number is incomparably greater for him who, recogniz- ing in the totality of phenomena the principal regulator of days and seasons, sees that nothing at the present moment can arrest the course of it. Buffon in his Political Arithmetic calculates differently the preceding probability. He supposes that it differs from unity only by a fraction whose numerator is unity and whose denominator is the number 2 raised to a power equal to the number of days which have elapsed since the epoch. But the true manner of relating past events with the probability of causes and of future events was unknown to this illustrious writer. ToC20 CHAPTER IV. CONCERNING HOPE. THE probability of events serves to determine the hope or the fear of persons interested in their exist- ence. The word hope has various acceptations; it expresses generally the advantage of that one who expects a certain benefit in suppositions which are only probable. This advantage in the theory of chance is a product of the sum hoped for by the probability of obtaining it; it is the partial sum which ought to result when we do not wish to run the risks of the event in supposing that the division is made proportional to the probabilities. This division is the only equitable one when all strange circumstances are eliminated; because an equal degree of probability gives an equal right to the sum hoped for. We will call this advantage mathematical hope. Eighth Principle. When the advantage depends on several events it is obtained by taking the sum of the products of the probability of each event by the benefit attached to its occurrence. Let us apply this principle to some examples. Let CONCERNING HOPE. 21 us suppose that at the play of heads and tails Paul receives two francs if he throws heads at the first throw and five francs if he throws it only at the second. Multiplying two francs by the probability of the first case, and five francs by the probability of the second case, the sum of the products, or two and a quarter francs, will be Paul's advantage. It is the sum which he ought to give in advance to that one who has given him this advantage; for, in order to maintain the equality of the play, the throw ought to be equal to the advantage which it procures. If Paul receives two francs by throwing heads at the first and five francs by throwing it at the second throw, whether he has thrown it or not at the first, the prob- ability of throwing heads at the second throw being , multiplying two francs and five francs by the sum of these products will give three and one half francs for Paul's advantage and consequently for his stake at the game. Ninth Principle. In a series of probable events of which the ones produce a benefit and the others a loss, we shall have the advantage which results from it by making a sum of the products of the probability of each favorable event by the benefit which it procures, and subtracting from this sum that of the products of the probability of each unfavorable event by the loss which is attached to it. If the second sum is greater than the first, the benefit becomes a loss and hope is changed to fear. Consequently we ought always in the conduct of life to make the product of the benefit hoped for, by its probability, at least equal to the similar product relative 22 A PHILOSOPHICAL ESSAY ON PROBABILITIES. to the loss. But it is necessary, in order to attain this, to appreciate exactly the advantages, the losses, and their respective probabilities. For this a great accuracy of mind, a delicate judgment, and a great experience in affairs is necessary ; it is necessary to know how to guard one's self against prejudices, illusions of fear or hope, and erroneous ideas, ideas of fortune and happi- ness, with which the majority of people feed their self- love. The application of the preceding principles to the following question has greatly exercised the geometri- cians. Paul plays at heads and tails with the condition of receiving two francs if he throws heads at the first thro\v, four francs if he throws it only at the second throw, eight francs if he throws it only at the third, and so on. His stake at the play ought to be, accord- ing to the eighth principle, equal to the number of throws, so that if the game continues to infinity the stake ought to be infinite. However, no reasonable man would wish to risk at this game even a small sum, for example five francs. Whence comes this differ- ence between the result of calculation and the indication of common sense ? We soon recognize that it amounts to this : that the moral advantage which a benefit pro- cures for us is not proportional to this benefit and that it depends upon a thousand circumstances, often very difficult to define, but of which the most general and most important is that of fortune. Indeed it is apparent that one franc has much greater value for him who possesses only a hundred than for a millionaire. We ought then to distinguish in the hoped-for benefit its absolute from its relative value. CONCERNING HOPE. 23 But the latter is regulated by the motives which make it desirable, whereas the first is independent of them. The general principle for appreciating this relative value cannot be given, but here is one proposed by Daniel Bernoulli which will serve in many cases. Tenth Principle. The relative value of an infinitely small sum is equal to its absolute value divided by the total benefit of the person interested. This supposes that every one has a certain benefit whose value can never be estimated as zero. Indeed even that one who possesses nothing always gives to the product of his labor and to his hopes a value at least equal to that which is absolutely necessary to sustain him. If we apply analysis to the principle just propounded, we obtain the following rule : Let us designate by unity the part of the fortune of an individual, independent of his expectations. If we determine the different values that this fortune may have by virtue of these expecta- tions and their probabilities, the product of these values raised respectively to the powers indicated by their probabilities will be the physical fortune which would procure for the individual the same moral advantage which he receives from the part of his fortune taken as unity and from his expectations ; by subtracting unity from the product, the difference will be the increase of the physical fortune due to expectations : we will call this increase moral hope. It is easy to see that it coin- cides with mathematical hope when the fortune taken as unity becomes infinite in reference to the variations which it receives from the expectations. But when these variations are an appreciable part of this unity 24 A PHILOSOPHICAL ESSAY ON PROBABILITIES. the two hopes may differ very materially among them- selves. This rule conduces to results conformable to the indications of common sense which can by this means be appreciated with some exactitude. Thus in the preceding question it is found that if the fortune of Paul is two hundred francs, he ought not reasonably to stake more than nine francs. The same rule leads us again to distribute the danger over several parts of a benefit expected rather than to expose the entire benefit to this danger. It results similarly that at the fairest game the loss is always greater than the gain. Let us suppose, for example, that a player having a fortune of one hundred francs risks fifty at the play of heads and tails; his fortune after his stake at the play will be reduced to eighty-seven francs, that is to say, this last sum would procure for the player the same moral advantage as the state of his fortune after the stake. The play is then disadvantageous even in the case where the stake is equal to the product of the sum hoped for, by its probability. We can judge by this of the immorality of games in which the sum hoped for is below this product. They subsist only by false reasonings and by the cupidity which they excite and which, leading the people to sacrifice their necessaries to chimerical hopes whose improbability they are not in condition to appreciate, are the source of an infinity of evils. The disadvantage of games of chance, the advantage of not exposing to the same danger the whole benefit that is expected, and all the similar results indicated by common sense, subsist, whatever may be the function CONCERNING HOPE. 25 of the physical fortune which for each individual expresses his moral fortune. It is enough that the proportion of the increase of this function to the increase of the physical fortune diminishes in the measure that the latter increases. ToC

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