Probability, every proposition considered in and of itself is true or false; but for us, it can be certain or uncertain; we can more or less perceive the relations which can exist between two ideas, or the agreement of one with the other, founded on certain conditions which link them, and which, when they are completely known to us, give us certainty of this truth, or of this proposition; but if we only understand a part of it, we then have only a simple probability , which seems that much more believable the greater the number of these conditions of which we are assured. It is these latter which form the degrees of probability, a correct opinion and an exact measurement of which would offer the pinnacle of wisdom and of prudence.

Geometers judged that their calculations could serve to evaluate these degrees of probability , at least to a certain point, and they had recourse to Logic, or the art of reasoning, to discover its principles and establish a theory for it. They regarded certainty as a whole and probabilities as parts of this whole. Consequently the correct degree of probability of a proposition was known to them exactly, when they could state and prove that this probability was worth a half, a quarter, or a third of this certainty. Often they were satisfied in supposing it; their calculations in and of themselves were no less correct; and these expressions, which at first can seem a bit bizarre, are no less meaningful. Examples taken from games, bets, or insurance will clarify them. Suppose someone comes and tells me that I have drawn the sum of ten thousand pounds in a lottery, I doubt the truth of this news. Someone who was there asks me what sum I would wager on the truth of that statement. I offer him half, which means that I only regard the probability of this news as a half-certainty; but if I had only offered a thousand pounds, it would have meant that I had nine times as much reason to believe in the truth of this news than not to believe in it. Or it would bring the probability to nine degrees, in the sense that certainty having ten, there would only lack one the addition of one more to give complete faith in the news.

In ordinary usage, we call probable what has more than a half-certainty of credibility, or what surpasses this half considerably; and morally certain , what is virtually complete certainty. We speak here only of moral certainty, which coincides with mathematical certainty, although it is not susceptible to the same proofs. Moral evidence is thus, properly speaking, a probability so great that it is wise for a man to think and act, in cases where one has this certainty, as one should think and act, if one had a mathematical certainty of it. There is moral evidence that there is a city of Rome: the contrary does not imply a contradiction; it is not impossible that all those who tell me that they have seen it, are in agreement to fool me, that the books which speak of it are made expressly for that purpose, that the monuments we have of it are only imagined; however, if I refused to yield to the evidence supported by the proofs I have of the evidence of Rome, simply because they are not susceptible to a mathematical demonstration, one could treat me, with reason, as an absurd person, since the probability that there is a city of Rome is so much stronger than the suspicion that there cannot be one, that one could scarcely express this difference by a number, or the value of its probability . This example suffices to render comprehensible moral evidence and its degrees which are so many probabilities . A half-certainty forms the uncertain , properly speaking, where the mind, finding equal reasons on both sides, does not know which judgment to make, which side to take. In this state of equilibrium, the slightest proof determines us; often one searches for it where there is neither reason nor wisdom to search for it; and as it is quite difficult, in many cases, where the opposing reasons approach very close to equality, to determine which are the ones which ought to carry the argument, the wisest men extend the point of uncertainty; they do not affix it only to that state where the mind is equally drawn to both sides by the weight of reason; but they carry it even further, to every situation sufficiently close to it that one could not perceive the inequality; it follows that the country of uncertainty is more or less vast, according as the lack of light, logic and courage is greater or less. It is narrower for those who are the wisest, or the least wise; for audacity restricts it more than prudence, by the boldness of its decisions. Below this half-certainty or the uncertain, are found suspicion and doubt , which end at the certainty of the falseness of a proposition. A matter is false by moral evidence, when the probability of its existence is so strongly inferior to the contrary probability , that there are ten thousand, or a hundred thousand against one to bet that it does not exist.

These, then, are the degrees of probability between two opposing evidences. Before researching its principles, it will not be out of place in an article where one is not satisfied with simple geometric calculations, to establish some general rules which are regularly observed by wise and prudent persons.

1. It is unreasonable to search for probabilities and be satisfied with them when one can succeed with evidence. One would consider as foolish a mathematician who, to prove a proposition in geometry, would have recourse, to opinions and to credibility, when he could produce a demonstration; or a judge who would prefer to guess, based on the past life of a criminal, whether he is guilty, rather than to listen to his confession whereby he admits his crime.

2. It is not sufficient to examine one or two of the proofs one can immediately put forth, one must weigh in the balance all those proofs that can come to our knowledge, and serve to discover the truth. If one asks what probability there is that a man 50 years of age dies within the year, it does not suffice to consider that in general, out of a hundred 50-year-old persons, three or four die within the year, and conclude that there are 96 to bet against 4, or 24 against one; one must also pay attention to this man’s temperament, the current state of his health, his lifestyle, the country he inhabits; all of which are circumstances which influence the duration of his life.

3. It is not enough to consider proofs serving to establish a truth, one must also examine those which combat it. One asks whether a person known and absent from his country for 25 years, from whom no news has been had, ought to be regarded as dead? On one side it is said that, despite all sorts of searches one has learned nothing; that as a traveler he could have been exposed to a thousand dangers, that a disease might have carried him off in a place where he was unknown; that if he were still alive, he would not have neglected to send news, especially assuming, as one must, that he would have an inheritance to collect, and other reasons that one could put forth. But, against these considerations, one opposes others that ought not be neglected. It is said that the man in question is indolent, and that on other occasions did not write, that perhaps his letters were lost, that he could find himself unable to write. All this suffices to show that in all things, one must weigh the proofs, the probabilities on each side, oppose them against each other, because a very probable proposition can be false, and that in the matter of probability , there is none so strong that it could not be combated and destroyed by an even stronger contrary one. Whence the opposition seen every day between men’s judgments. Whence the majority of disputes which would end well, if we would not regard as evident what is only probable, and listen and weigh the reasons offered against our opinion.

4. Is it necessary to warn that in our judgments it is prudent only to acquiesce in any proposition in proportion to its degree of credibility? Whoever could observe this general rule, would have all the equity of spirit, all the prudence, all the wisdom possible. But how far we are from that! The most common minds can, with attention, discern the true from the false; others, having more penetration, know how to distinguish the probable from the uncertain or the doubtful; but only geniuses distinguished by their finesse can assign to each proposition its correct degree of credibility, and adjust his assent to it accordingly: ah but these geniuses are rare!

5. Furthermore, the wise and prudent man will not consider only the probability of success, he will also weigh the measure of the good or bad that can be expected in taking such a side, or in deciding in favor of the contrary, or in staying inactive; he will even prefer the side where he knows the appearance of success is quite slight, when he sees at the same time that the risk he runs is nothing or practically so; and that on the contrary if he succeeds, he can obtain a very considerable reward.

6. Since it is not possible to fix with the precision one would desire the degrees of probability , let us be satisfied with the close approximations that we can obtain. Sometimes, by a fastidiousness that is out of place, we expose ourselves and society to worse evils than those we were trying to avoid; it is an art to know how to pull away from perfection in certain situations, in order to approach it more closely in others which are more essential and more interesting.

7. Finally, it seems superfluous to add here that in uncertainty one ought to defer deciding and acting until one has more light on the situation, but that if the case is such as to permit no delay, one must decide in favor of what is most probable; and once the position that we have judged the wisest is taken, one must not repent of it, even when the outcome does not correspond to what we reasonably expected of it. If, in a fire, one cannot escape except by jumping out the window, one must decide in favor of this plan, however bad it is. Uncertainty would be far worse, and whatever the outcome, we have taken the wisest course, there must be no regrets.

After these general rules, the application of which will be easy to make, let us come to the sources of probability . We reduce them to two types: one includes the probabilities drawn from the consideration of nature itself, and of the number of causes or reasons which can influence the truthfulness of the proposition in question; the other is founded only on the experience in the past which can make us confidently draw conclusions for the future, at least when we are assured that the same causes which produced the past still exist, and are ready to produce the future.

An example will give us a better understanding of the nature of, and the difference between, these two sources of probability . I suppose that it is known that one has put thirty thousand bills in an urn, among which there are ten thousand black ones and twenty thousand white ones, and then one asks, what the probability is that in drawing one of them at random, it will turn out white? I say that by the simple consideration of the nature of things, and by comparing the number of causes which can make a white bill come out with the number of those which will make a black one come out, by that alone it is two times more probable that a white bill will come out than a black, in such a way that, as the bill which will come out is necessarily either white or black, if we divide this certainty into three equal degrees or parts, one will say that there are two degrees of probability of drawing a white bill, and one degree for the black bill, or that the probability of a white is 2/3 of certainty, and that of the black bill 1/3 of this certainty.

But suppose that I only see in the urn a large number of bills, without knowing the proportion that there is of whites to blacks, or even without knowing if there are not at all some of a third color, in this case how to determine the probability of drawing from it a white one? I say that it will be by making some trials, that is, by drawing one bill to see what it will be, then putting it back in the urn, then drawing a second which I put back as well, then a third, a fourth, and so on in sequence as long as I like. It is clear that the first bill drawn having come out white, gives only a very slight probability that the number of whites surpasses the number of blacks, a second white drawn would augment this probability, a third would fortify it. Indeed if I were to draw in sequence a large number of whites, I would in my right conclude that they are all white, and so conclude with increasing credibility the more bills I had drawn. But if, out of the first three bills, I draw two whites and one black, I can say that there is some very slight probability that there are two times more whites than blacks. If out of six bills four whites and two blacks come out, the probability grows, and it will grow as the number of trials or experiments confirm to me always the same proportion of whites to blacks. If I had made three thousand trials, and had two thousand white bills against a thousand black ones, I could scarcely doubt that there are two times more whites than blacks, and consequently that the probability of drawing a white is double that of drawing a black.

This manner of determining the relation of the causes which give rise to an event with those which make it fail to occur, or more generally the proportion of reasons or conditions which establish the truthfulness of a proposition with those which give the contrary, can be applied to anything that can happen or not happen, to everything which can be or not be. When I see on mortuary registers that during twenty, fifty or a hundred years, out of the number of babies born, one third die before the age of six years, I will conclude concerning a newborn that the probability it will live at least to the age of six years is 2/3 of certainty. If I see that of two players who play with equal marbles, the first always wins two matches, while the other only wins one, I will conclude with a great deal of probability that he is two times stronger than his opponent; if I remark that someone has lied to me on ten occasions out of the hundred that he spoke to me, the probability of his testimony will be in my mind only 9/10 of certainty, or even less.

The attention given to the past, the fidelity of memory for retaining what has happened, and the precision of registers for preserving events, constitute what the world calls experience . A man who has some experience is one who, having seen much and reflected much, can tell you approximately (for we are not going here to mathematical precision) what probability there is that a certain event having arrived, a certain other one will follow it; in this way, all other things being equal, the more tests or experiences one has had, the more one can be sure of the precise relation of the number of favorable causes to the number of contrary causes.

One could ask if this probability , increasing by a sequence of repeated experiments, can in the infinite become a moral certainty; or if these increases are so limited, that gradually diminishing they amount in the infinite only to a finite probability . For it is known that there are augmentations which, although perpetual, only create in the infinite a finite sum; for example, if the first experiment were to give a probability that was only a 1/3 of certainty, and the second a probability that was only a third of this third, and the third a probability that was only a third of the second, and a fourth a probability that was only a third of the third, and so on infinitely often. It would be easy to see by calculation that all these probabilities together only lead to a half-certainty, so that one would make an infinite number of experiments in vain, one would never come to a probability that would be confounded with moral certitude, which would lead to the conclusion that experience is useless, and that the past has proved nothing for the future.

Mr. Bernoulli, the geometer who best understood these sorts of calculations, proposed himself the objection and gave a response to it. It is found in his book de arte conjectandi, part 4 , in its full extent; a problem, according to him, as difficult as the quadrature of the circle. He shows there that the probability which was born from a repeated experiment was always increasing, and increasing in such a way that it was approaching indefinitely close to certainty. His calculation teaches us how to determine (the question having been proposed in a fixed manner) how many times one would have to repeat the experiment to attain an assigned degree of probability . Thus, in the case of an urn full of a large number of white and black balls, one wishes to assure oneself by experiment of the proportion of whites to blacks; Mr. Bernoulli finds that for it to be a thousand times more probable that there are two blacks for three whites than another given supposition, it will be necessary to have drawn from the urn 25550 balls, and that, for that to be ten thousand times more probable, it was necessary to have made 31258 trials, at last, in order that that became a hundred thousand times more probable, one would need 36966 drawings. The difficulty and length of the calculation do not permit reporting it here in its entirety, one can see it in the book cited.

It is thereby demonstrated that experience of the past is a principle of probability for the future; that we have reason to expect that events will conform to those that we have seen happen; and that the more frequently we have seen them arrive, the more we have reason to expect them anew. Once this principle is received, one senses how useful would be, in questions of Physics, Politics, and even regarding everyday life, exact tables which would fix upon a long sequence of events the proportion of those which arrive in a certain manner to those that arrive in a different one. The usage drawn from baptismal and mortuary registers is so great, that one should encourage not only perfecting them by marking, for example, age, condition, temperament, manner of death, etc., but also by making them for many other events that one calls quite inappropriately chance effects; in this way one could form tables marking how many fires arrive in a certain time, how many epidemic diseases make themselves felt in certain spaces of time, how many ships perish, etc., which would become very handy for resolving an infinity of useful questions, and would give attentive young people the experience of old folks.

It is of course clear that one will not give in to the abuse, which is too common, of the proof of experience, that one will not establish on the basis of a small number of facts a large probability , that one will not go so far as to oppose or even prefer a feeble probability to a contrary certainty, that one will not give in to the weakness of those players who only take the cards which won or which have lost, although it is evident by the nature of games of chance, that the preceding plays do not influence the following ones. A superstition however more pardonable than so many others which, on the basis of the slightest experience or least consequential reasoning, are only too much introduced into the flow of life.

To these two general principles of probability , we can join other more specific ones, such as the equal possibility of many events, the understanding of causes, testimony, analogy and hypotheses .

1.When we are assured that a certain thing can only happen in a certain determined number of ways, and we know or suppose that all these ways have an equal possibility, we can say with assurance that the probability it will happen in such-and-such a manner, is worth, or is equal to, that many parts of certainty. I know, for example, that in throwing a die at random, I am sure to roll either the 1 point, or the 2, or the 3, or the 4, or the 5, or the 6. Let us suppose moreover that the die is perfectly fair, the possibility is the same for all the points. There are then six equal probabilities here, which all together form certainty; in this way each one is a sixth part of certainty. This principle, as simple as it seems, is infinitely fruitful; it is on the basis of it that are framed all the calculations that have been made, and that one can render the probabilities concerning games of chance, lotteries, insurance, and in general concerning anything at all, susceptible to calculation. It is only a matter of great patience and a detailed list of the combinations, to determine the number of favorable events and the number of contrary ones. It is on the basis of this principle, joined to experience, that one determines the probabilities of human life, or of the time that a person of a certain age can reasonably flatter himself that he will live; which gives the foundation for the calculation of the value of annuities or tontines. See the essays on probabilities of human life, and the works cited at the end of this article. It extends to the calculation of annuities put upon two or three persons payable to the last survivor, on benefits, alimentary pensions, insurance contracts, bets, etc.

I said that this principle is employed when we suppose the various cases equally possible. And in fact, it is only by supposition relative to our limited knowledge that we say, for example, that all the points of a die can equally arise; it is not that when they roll in the dice-box the one that is supposed to turn up does not already have the disposition which, combined with that of the dice-box, the table, or the force and the manner that one tosses the die, makes it sure to arrive; but all that being entirely unknown to us, we do not have any reason to prefer one point to another; we thus suppose them all equally easy to roll. However there can often be some error in this supposition. If one wanted to find the probability of rolling 8 points with two dice, one would commit a great sophism to reason as follows: with two dice, I can roll a 2, or 3, or 4, or 5, or 6, or 7, or 8, or 9, or 10, or 11, or 12 points; thus the probability of rolling 8, will be 1/11 of certainty; for this would suppose that these 11 points would be equally easy to roll, which is not true. The simplest calculations on the game of tric-trac teach us that out of the 36 equally likely throws with two dice, 5 give us the point of 8; the probability will thus be 5 out of 36, or 5/36 of certainty, and not 1/11.

This sophism is easily avoided in calculations concerning games, where it is easy to determine the equal or unequal possibility of events; but it is more hidden, and only too common in the most composite cases. Thus many people complain that they are quite unfortunate, because they have not been able to obtain a certain boon which fell to others as their share; they suppose that it was equally possible, equally convenient, that this benefit come to them, without wanting to consider that they were not in as advantageous a position, that they had for themselves only one favorable possibility, while the others had many of them, so that it would have been extremely lucky that this sole possibility happen, without saying that the events which we attribute to chance are directed by an infinitely wise providence, which has calculated everything, and which, for reasons unknown to us, disposes things in a manner much more convenient than the arrangement that our feeble lights or our passions would wish to impose there.

Following simple probability comes compound probability , which depends again on the same principle. It’s the probability of an event that cannot happen except in the case that another event, itself simple, happens. An example will explain it. I suppose that in a game of quadrille with 40 cards I am asked to draw a heart, the probability of succeeding is 1/4 of certainty, since there are 4 colors and 10 equally possible cards of each color. But if I am told that next that I will win if I draw the king of hearts, then the probability becomes composite; for 1. I must draw a heart, and the probability is 1/4: 2. assuming that I have drawn a heart, the probability will be 1/10, since there are 9 other hearts that I can just as well draw as the king. This probability grafted onto the first is but the tenth of a quarter, or 1/4 of 1/10, that is, 1/40 of certainty. And it is clear that since out of 40 cards I am to draw precisely the king of hearts, I only have one favorable case out of 40 equally possible ones, or one favorable against 39.

This compound probability is thus estimated by taking from the first part that which one would take from entire certainty, if this probability were converted to certainty. A friend left for the Indies on a fleet of twelve vessels; I learn that three of them have perished, and that a third of those personnel that were saved died on the voyage; the probability that my friend is on one of those vessels that made it to port is 9/12, and that that he is not among the third who died en route is 2/3. The compound probability that he is still alive, will thus be 2/3 of 9/12 or 6/12, or a half-certainty. He is therefore, for me, between life and death.

This calculation can be applied to all sorts of proofs or reasonings, reduced for greater clarity to the form prescribed by the art of reasoning: if one of the premises is certain, and the other probable, the conclusion will have the same degree of probability as this latter premise; but if one and the other are simply probable, the conclusion will only have a probability of probability , which is measured by taking from the probability of the greater such a part as is expressed by the fraction which measures the probability of the lesser.

In these last examples the 9/12 of 2/3, which is the probability of the greater, and thus the value of the conclusion, will be 6/12 or 1/2.

Whence it appears that the probability of a probability only creates a very slight probability . What will be a probability of the third or fourth degree? Or what is there to think of those reasonings which are so frequent, whose conclusion is based only on several probable propositions which all have to be true for the conclusion to be as well? But if it sufficed that one of them held to verify the conclusion, it would be quite the opposite; the more one piles up probabilities , the more probable the matter would become. If, for example, someone told me, I will give you a louis if you roll two dice and get 8 points, the probability of rolling 8 is 5/36; if he added, I’ll give it to you also if you roll a 6: then, to win, it suffices to roll one or the other, my probability would be 5/36 and 5/36, that is, 10/36, which augments my hope of winning.

So much for the elements based on which one can determine all questions, and the examples depending on this first principle of probability .

2. Let us pass to the second, which is knowledge of causes and signs, which one can regard as causes or random effects. We will only make one particular comment on probabilities , referring the reader to the article Cause. There are causes whose existence is certain, but whose effect is but doubtful or probable; there are others whose effect is certain, but whose existence is doubtful; finally, there can be others whose existence and effect are only a simple probability . This distinction is necessary; an example will explain it. A friend has not responded to my letter; I seek the cause, three come to mind: he is lazy, maybe he is dead, or his work prevents him from responding to me. He is lazy, a first cause whose existence is certain: I know that he writes with great difficulty; but the effect of this cause is uncertain, because a lazy man does decide sometimes to write. He is dead, a second cause which is very uncertain, but whose effect would very well be certain. He has work to do, a third cause which is uncertain in and of itself: I suspect only that he has a lot of work, and even assuming the existence of it, the effect would still be uncertain, since one can have work and yet still find the time to write.

The same thing applies to signs; their existence can be doubtful, their meaning uncertain; and the existence and meaning can have only credibility. The barometer descends, it’s a sign of rain whose existence is certain, but whose meaning is doubtful; often the barometer descends without rain.

From this distinction it follows that the conclusion drawn from a cause or sign whose existence is certain, has the same degree of probability that is found in the effect of this cause, or in the meaning of this sign. We have only to reduce the example of the barometer to this form. If the barometer descends, we will have rain: that is only probable; but the barometer descends, that is certain: thus we will have rain; a probable conclusion, whose value derives from experience. Similarly if the existence of the cause or sign is doubtful, but its effect or meaning is not, the conclusion will have the same degree of probability as the existence of the cause or sign. My friend being dead, that is doubtful; the conclusion that I draw from it, that he cannot write me, will be equally doubtful.

But when the existence and the effect of the cause are probable, or where it concerns signs when the existence and meaning of the sign are but doubtful, then the conclusion will only have a compound probability . Let us suppose that the probability that my friend has work is 3/4 of certainty, and that the probability that this work, if he has it to do, prevents him from writing me is 2/3 of this certainty, then the probability that he will not write me will be composed of the two others, which will be a half-certainty.

3. We have indicated testimony as a third principle of probability ; and it relates so closely to the subject whose principles we are giving that we cannot dispense with reporting here what there is to say relative to probabilities and moral certainty. We cannot see everything by ourselves; there are infinitely many things, often the most interesting, for which one must depend on the testimony of others. It is therefore important to determine, if not with exactness, at least in a manner approaching it, the degree of assent that we can give to this testimony, and what is for us its probability .

When someone tells us a story, or advances a proposition from the number of those which are proved by witnesses, one ought to examine first the very nature of the matter, and then weigh the authority of the witnesses. If in both of these, one finds that none of the conditions required for the truth of the proposition lack, one cannot refuse to acquiesce to it; if it is evident that one or more of these conditions are lacking, one ought not hesitate to reject it; finally, if one sees clearly the existence of some of these conditions, but remains uncertain about the others, the proposition will be probable, and all the more so as a greater number of these conditions hold.

1. As to the nature of the matter, the only condition required, is that it be possible, that is, that there be nothing in nature preventing its existence, and consequently nothing which would prevent me from believing it as soon as it is sufficiently proved by an external proof, such as that given by testimony. On the other hand, if the thing is impossible, if it has in and of itself an invincible repugnance for existing, no matter what degree of credibility proofs via testimony might offer elsewhere, or other extrinsic reasons for its existence, I could not believe it. Should someone advance a contradiction, an absolute impossibility, joining to it all sorts of proofs, he would never succeed in persuading me of what is metaphysically impossible. A squared circle can neither be understood nor accepted. Does it concern a physical impossibility? We will be a bit less difficult; we know that God himself has established the laws of nature, that he is constant in the observation of these laws; thus the mind revolts at the idea that they can be violated. However we know that he who has established them has the power to suspend them; that they are not absolutely necessary, but only convenient. Thus we ought not absolutely refuse to put confidence in witnesses or in external proofs to the contrary; but these proofs must be quite evident, quite numerous, and presented properly and appropriately for us to accept them. Is it a question of a moral impossibility or of an opposition to the moral qualities of intelligent beings? Although far less delicate concerning the proofs or witnesses who want to persuade us of it, nevertheless we must see in them that credibility which is found in the their very character, and in the effects which result from it; actions must necessarily follow from the principles which ordinarily produce them: in this way it seems impossible that a wise man, of grave and modest character, would betake himself without motive or reason to commit an indecency in public. On the other hand, of an ordinary morally possible fact, in conformance with the course of the law of nature, we are easily persuaded; it carries within itself many degrees of probability ; whatever little testimony may add, it becomes very probable. This probability will increase further with the agreement of one truth with others already known and established; if the story we are told is carefully tied to history, if we would not be able to deny it without overturning a sequence of verified historical facts, just by that it is proven; if on the contrary it cannot be placed in history without upsetting certain known great events, just for that this story is rejected. Why do we regard the history of the Greeks and Romans as much more believable than that of the Chinese? It’s because we have an infinity of monuments of every kind which have a relationship so necessary, or at least so natural, with this history, and which so link them to history in general, that they multiply its proofs out to infinity; whereas that of the Chinese has but few connections with the course of this general history as we know it.

2. When the proofs drawn from the very nature of the thing have been weighed, and the possibility and in some manner the degree of intrinsic probability recognized, one must turn to the validity of the testimony itself. It depends on two things, the number of witnesses, and the confidence one can have in each one of them.

For the number of witnesses, there is no one who would not sense that their testimony is all the more probable, when they are in greater number: one would even believe that its probability increases in the same proportion as the number; so that two witnesses worthy of equal confidence would make a probability double that of one sole one; but one would be mistaken. The probability grows with the number of witnesses in a different proportion. If we suppose that the first witness gives me a probability amounting to 9/10 of certainty, the second one, whom I suppose equally credible, would he add to the probability of the first another 9/10? No, since their two testimonies combined would make 18/10 of certainty, or a certainty plus 8/10 more, which is impossible. I say then that this second witness will increase the probability of first one by 9/10 out of what remains to go to certainty, and thus will push the combined probability to 99/100, a third one will bring it to 999/1000, a fourth to 9999/1000, and so on in sequence, approaching ever closer to certainty, without ever arriving there entirely: which ought not surprise, since however many witnesses we suppose, there must always remain the contrary possibility, or some degrees of probability (very slight, it is true) that they are mistaken: here is the proof. When two witnesses say a thing, it must be that, for me to be mistaken in believing their testimony, the one and the other both lead me into error; if I am sure of one of them, it is of little importance that the other be credible. Now the probability that they both mislead me, is a probability composed of two probabilities, that the first one misleads, and that the second one misleads. That of the first is 1/10 (since the probability that the thing conforms to his testimony is 9/10); the probability that the second misleads me as well, is also 1/10; thus the probability of the contrary, that is, that one or the other is telling the truth, is 99/100.

One sees that here I am representing moral certainty as the end of a course that the various witnesses who come to the support of each other make me run. The first one makes me move a distance which is in the same proportion with the whole course as the force of his testimony is with entire certainty. If his report produces in me 9/10 of certainty, this first witness makes me move 9/10 of the way. A second witness comes, just as credible as the first; he advances me, out of the way that remains, precisely as much as the first one made me advance out of the total space; this latter had led me across 9/10 of the course, the second makes approach still further across 9/10 of that tenth which remained; so that with these two witnesses I’ve accomplished 99/100 of the whole. A third one with the same weight makes me run 9/10 of the hundredth remaining, between certainty and the point where I am; there will only remain the thousandth part, and I would have run 999/1000 of the course, and so on in sequence.

This method of calculating the probability of testimony, is the same for a number of witnesses whose credibility is different; which ordinarily conforms more to the nature of things. Let an action be reported by three witnesses; the report of the first is equivalent to 5/6 of certainty, the second only produces 2/3, and the third, less credible than the other two, would only give me 1/2 certainty if he were alone. Then, assuming I have no reason to suspect any coordinating among them, I say that their combined testimony gives me a probability which is 35/36 of certainty, because the first advancing 5/6, there remains 1/6, of which the second will make me run 2/3; in this way there will still remain 1/3 of 1/6, which is 1/18; and the third one advancing me half, I am no longer further from the end of the course than 1/2; I would then have run 35/36; moreover, it does not matter in which order I take them, the result is the same.

2. This principle can suffice for calculations of the value of testimony. As to the trust each witness deserves, this is founded upon his capacity and his integrity. By the first he cannot be mistaken, by the second he is not seeking to mislead: two equally necessary conditions; one without the other will not suffice. Whence it follows that the probability inspired by the report of a witness in whom we recognize this capacity and this integrity, ought to be regarded and calculated as a compound probability. A man comes and tells me that I have won the lottery; I know that he is not too intelligent; he can be mistaken: all things considered, I evaluate the probability of his capacity at 8/9; but perhaps he would enjoy misleading me. Suppose that there are 15 to wager against 1 that he is acting in good faith; the probability of his integrity will then be 15/16. I say that the safety of his testimony, or the probability compounded by his capacity and his integrity, will be 8/9 of 15/16, that is 5/6 of certainty.

The surest way to judge the capacity and the integrity of a witness, would be experience . We would have to know exactly how many times this man misled or told the truth; but this experience is limited, and ordinarily is lacking. Instead one has recourse to public and private reports, in external circumstances in which this witness is involved. Has he received a good education? Is he of a rank which is supposed to engage him to respect the truth more? Is he of an age that gives more weight to his testimony? Is he disinterested in the matter? Or what can be his goal? Does he obtain some advantage from it? Or does he avoid some penalty? Does it flatter his taste and passion to mislead us? Is it a consequence of prejudice, of hate? All of these are circumstances that one must examine if we do not have experience, and whose exact value is difficult to determine.

Furthermore, the capacity of a witness supposes, aside from well-tuned senses, a certain strength of spirit which is not frightened by danger, nor surprised by novelty, nor carried away by too precipitous a judgment. He is more credible in proportion as the matter of which he speaks is more familiar and better-known to him; his story itself often proves his capacity, and tells me that he has taken or neglected all the necessary precautions not to be mistaken: the more often he has reiterated it, the greater is his right to my confidence. This capacity for good recognition depends further on his attention in observing, his memory, the time; other conditions which, joined to the manner of narrating clearly and in detail, influence the degree of probability which the witness will earn.

One should not neglect the silence of those who would have an interest in contradicting a testimony, at least if it is extorted neither by fear nor by authority. In truth it is difficult to evaluate the weight of such a negative testimony; one can assure in general that he who simply holds silent, deserves less attention than he who offers assurance for a fact. If nevertheless the fact is such that he could not have failed to know it, if it would have served to validate the rest of his story, if he would have had an interest in reporting it, or if his duty would have called him to it; in cases like these it is certain that his silence is a worthy testimony, or at least it enfeebles and diminishes the probability of opposing testimonies.

We still have to say a word about testimonies by hearsay, or about the enfeeblement of a testimony which, passing from mouth to mouth, only makes its way to us via a chain of witnesses. It is clear that a hearsay witness, all other things being equal, is less credible than an eyewitness; for if the latter is mistaken or wanted to mislead, the hearsay witness who follows, although faithful, will only report an error to us; and even when the first would have told the truth, if the hearsay witness is not faithful, if he misunderstood, if he forgot or confused some essential part of the story, if he mixes in some of his own, he no longer reports to us the pure truth; in this way the confidence we owe this second testimony is already enfeebled, and will be enfeebled according as it passes by more mouths, according as the chain of witnesses is extended. It is easy to calculate, on the basis of established principles, the proportion of this enfeeblement.

Let us follow the example we have used. Pierre tells me that I have won a lottery of a thousand pounds: I estimate his testimony at 9/10 of certainty, that is, I hold my expectation at 900 francs. But Pierre tells me that he knows it from Jacques; now, if Jacques had spoken to me of it, I would have estimated his report at 9/10 in supposing him as credible as Pierre; thus, not entirely sure that Pierre didn’t make some mistake in receiving this testimony from Jacques, or that he does not have some design to fool me, I ought only count on 9/10 of 900 pounds, or on 9/10 of 9/10 of 1000 pounds, which makes 810 pounds. If Jacques got this fact from another, I should take from this last assurance 9/10, assuming this third equally credible, and my expectation would be reduced to 9/10 of 9/10 of 9/10 of 1000 pounds, or at 729 pounds, and so on in sequence.

Whoever takes the trouble to calculate using this method will find that if the confidence that we ought to have in each witness is 95/100, the thirteenth witness will not transmit more than 1/2 certainty, and then the matter will cease to be probable, or there will no longer be any external reason for believing it. If the probability due each witness is 99/100, it is only reduced to 1/2 certainty when the testimony has passed by seventy mouths; and if this confidence is assumed to be 999/1000, we would need a chain of 700 witnesses to render the fact uncertain.

These quite long calculations can be abridged by this general rule, for which simple algebra furnishes us the result and demonstration. Take 7/10 of the quotient of the division of the probability of simple witness by the contrary probability , as here would be 95/100 by 5/100, or 95 by 5, which is 19, of which I take 7/10, and you will have the witness who leaves you with a half-certainty; in this example it’s 13 3/10, which gives the thirteenth witness.

It will be the same if the successive witnesses are assumed to be of unequal force; whence one can reasonably conjecture in general, that we must make little case of hearsays, without however letting ourselves fall into an historical pyrrhonism, since here one can combine the probabilities given by several collateral chains of successive witnesses. Suppose a fact comes to us by a simple succession of live-voice witnesses, in such a way that each witness succeeds the other at the end of twenty years, and that the confidence in each witness diminishes by 1/20; by the preceding rule, at the end of twelve successions, or of 240 years, the fact will become uncertain, if only proved by these 12 witnesses; but if this chain if witnesses is fortified by nine other similar chains which agree in attesting to the same truth, then there will be more than a thousand to wager against one for the truth of the fact; if we assume a hundred chains of witnesses, there will be more than two million against one in favor of the fact.

If the testimony is transmitted in writing, the probability augments infinitely, inasmuch as it subsists and is preserved for a much longer time; the concordant testimony of many copies or printed books which form so many different chains, gives a probability so great that it approaches indefinitely close to certainty; for in supposing that each copy can endure 100 years, which is at the very least, and that at the end of this term the authority, not only of a sole copy, but of all those that have been made of the same original, is 99/100, then we would have to have more than seventy successions of 100 years, or 7,000 years, for the fact to become uncertain; and if we suppose several chains of witnesses, who agree in attesting to the same fact, the probability augments so strongly that it becomes infinitely little different from entire certainty, and will far surpass the assurance that one could have from the mouth of one or even of many eyewitnesses. There are other circumstances that are easy to suppose and which demonstrate the great superiority of written tradition over oral tradition.

We have indicated two other sources of probability , for the analogy and hypotheses concerning which we refer to the articles Induction, Analogy, Hypothesis, Supposition. These principles can suffice for explaining the whole theory of probability . We have only given the elements; one will find their applications in all the good works, which are in great number, on this subject. Such are the Essais sur les probabilités de la vie humaine , by Mr. Deparcieu; l’Analyse des jeux de hasard , by Mr. de Montmord, which gives the theory of combinations, as well as the article in this Dictionary under this word, and many others which refer to it, above all the Ars conjectandi , by Mr. Jacq. Bernoulli, and the Mémoires of Mr. Halley, which are found in the Transactions of England, n. 196 and following, which all serve to determine the credibility of events, and the degrees by which we arrive at moral certainty.

Let us conclude that it would not be entirely impossible to reduce all this theory of probabilities to a quite formalized calculation, if some good talents would compete, by research, observations, an extended study, and an analysis of heart and mind, founded on experience, at cultivating this branch of knowledge which is so important, and so useful in the continual practice of life. We concur that there is still much to do, but consideration of what lacks ought to excite people to fill these voids, and the importance of the subject offers enough to amply reward the difficulties.