Philosophical Essay on Probabilities

All events, even those that on account of their rarity {or insignificance} seem not to obey the great laws of nature, are as necessary a consequence of these laws as the revolutions of the sun. Ignorant of the bonds that link them to the entire system of the universe, we have made them depend on final causes, or on chance, according as they occur and succeed each other in a regular fashion, or without apparent order. But these fancied causes have been successively moved back as the boundaries of our knowledge have expanded, and they vanish entirely in the face of a sound philosophy, which sees in them only the expression of our ignorance of the true causes.

The first of these principles is the very definition of probability, which, as we have seen, is the ratio of the number of favourable cases to that of all possible cases [1].

The probability of events may be useful in determining the hope [1] or fear of people affected by their occurrence. The word expectation has various interpretations: in general it expresses the advantage to anyone who expects any benefit whatsoever, under assumptions that are only probable. This advantage, in the theory of chances, is the product of the sum expected and the probability of getting it; it is the partial sum that ought to be paid out when one does not want to rim the risks of the event, supposing that the total sum is to be shared out in proportion to the probabilities. This allocation is the only fair one when all irrelevant circumstances are set aside, because an equal degree of probability gives an equal claim to the sum expected. We shall call this advantage mathematical expectation [2].

In the search for methods called for by the application of the principles we have just discussed to various questions in probability, several branches of Analysis, in particular the theory of combinations and the calculus of finite differences, have arisen [1].

Early research in probability was concerned with the combinations arising in games of chance. Among the infinite variety of these combinations, some are easy to calculate, while others cause more difficulty, and since the difficulties increase as the combinations become more complicated, curiosity and the desire to overcome these difficulties prompted mathematicians to improve more and more upon this kind of analysis [1]. We have already seen that the profits shown by a lottery can easily be determined by the theory of combinations. But it is more difficult to know how many draws are needed for one to be able to lay a bet of 1 to 1, for example, that all numbers will be drawn. If n is the number of numbers, r the number of numbers drawn on each draw, and i the unknown number of draws, the expression for the probability that all the numbers will be drawn will depend on the nth finite difference of the ith power of the product of r consecutive numbers [2]. When n is very large, it becomes impossible to find the value of i that makes this probability equal to 1/2, unless this {finite} difference is converted into a series that converges rapidly. This can be successfully carried out by the method indicated above for approximating functions of very large numbers {of observations}. Thus in a lottery [3] composed of 10,000 numbers, only one of which is drawn at a time, there is some disadvantage in a 1 to 1 bet that all the numbers will appear in 95,767 draws, and an advantage in laying the same bet on 95,768 draws.

Inequalities of this kind have an obvious influence on the results of calculations of probabilities, an influence that deserves particular attention [1]. Consider the game of heads or tails, and suppose that it is equally easy to get one or the other face when the coin is tossed. Then the probability of getting heads on the first toss is 1/2, and that of getting it twice running is 1/4. But if the coin is biased in favour of one of the faces rather than the other, and if one does not know which face is favoured by this bias, the probability of getting heads on the first toss will still be 1/2, because, if one is ignorant of the face favoured by the bias, the probability of the simple event is increased or decreased by the same amount depending on whether or no the bias is in its favour. But even in this state of ignorance the probability of getting heads twice running is increased. Indeed, this probability is found by multiplying that of getting heads on the first toss by the probability that, having thrown heads on the first toss, one will throw it on the second. Now the occurrence of heads on the first toss gives reason to believe that the coin is biased in favour of this side. Consequently in such a case the unknown bias increases the probability of getting heads on the second toss, and as a result it increases the product of these two probabilities. As a numerical example, let us suppose that this bias increases the probability of the favoured simple event by a twentieth.

Among the variable and unknown causes that are included under the name of chance, and that make the course of events uncertain and irregular, a striking regularity is seen to arise as these events increase in number, a regularity that seems to follow a plan, and that has been considered as evidence of Providence ^α [1]. But if we think about this, we soon realize that this regularity is only the {natural} progression of the respective possibilities of simple events; the more probable these events are, the more often they ought to occur. Imagine, for example, an urn containing white and black balls, and suppose that each time a ball is drawn it is replaced in the urn before a new draw is made. The ratio of the number of white balls drawn to the number of black balls drawn will most often vary considerably in the first few draws: but the variable causes of this irregularity produce results alternately favourable and contrary to the regular course of events — results that, cancelling each other out in the ensemble of a large number of draws, allow better and better estimation of the ratio of the white to the black balls contained in the urn, or of the respective possibilities of getting a white or a black ball on each draw. This leads to the following theorem:

The probability that the ratio of the number of white balls drawn to the total number of balls drawn does not differ ^a from the ratio of the number of white balls to the total number of balls in the urn ^a by more than a given amount, tends to certainty as the number of events keeps on increasing, no matter how small this given amount may be [2].

Natural phenomena are most often surrounded by so many foreign circumstances, and so many perturbing causes confound their influence, that it is very difficult ^α to recognize them [1]. In such a case they can be discovered only by increasing the number of observations ^a or experiences {or experiments} ^a , so that as the foreign effects finally cancel each other out, these phenomena ^b and their various components ^b are clearly revealed by the mean results. ^{β c} The more observations there are, and the less they differ from one another, the more their results approach the truth. This latter condition can be realized by the choice of the methods of observation, by the precision of the instruments, and by taking care to make accurate observations. Then the optimal {or most advantageous} mean results, or those that give the smallest error, are determined by probability theory. But this is not enough: it is moreover necessary to estimate the probability that the errors in these results lie within given limits, for without this one would have only an imperfect knowledge of the degree of accuracy obtained. Formulae appropriate to the achieving of these ends are thus a true improvement of the scientific method, and it is very important that they be subjoined to that method.

^a We have just seen the advantages of probabilistic analysis in the search for laws of natural phenomena whose causes are either unknown or so complicated that their effects cannot be submitted to the calculus. This is the case in almost all topics in the moral sciences [1]. ^a So many unforeseen causes, either concealed or inappreciable, affect human institutions that it is impossible to determine their consequences a priori. ^b The sequence of events that time brings about explains these consequences ^b , and shows how to remedy those that are prejudicial. Wise laws have often been enacted in this respect; but because we have neglected to preserve the reasons for them, several have been repealed as useless, and for their reinstatement it is necessary that unfortunate experiences again show the need for them. Hence, it is very important, in each branch of public administration, to keep an exact record of the results that have been produced by the various methods that have been used, ^c and that are so many experiments made on a large scale by governments ^c . Let us apply to the political and moral sciences the method based on observation and the calculus, a method that has served us so successfully in the natural sciences[2].

The search for these means is very important in natural philosophy, and the analysis that it requires is the most delicate and the most intricate in the whole of probability theory. The most precise observations and experiments are always subject to errors that influence the values of the elements one wishes to infer from them. To get rid of these errors, in as much as it is possible, by doing away with them one by one, we must increase the number of observations; for the more observations there are, the more accurate will the mean be. But what is the best way to form this mean? Of what error is this result still susceptible? Only probabilistic analysis can answer these questions; and this is what it teaches us.

^a Since most of our opinions are founded on the probability of testimony {or evidence}, it is very important to submit such evidence to the calculus [1]. Very often, it is true, things become impossible, because of the difficulty of estimating the veracity of witnesses and because of the large numbers of circumstances that accompany the facts they attest. But in many cases one can solve problems that are closely analogous to the questions proposed, the solutions being able to be regarded as fitting approximations to guide us and to guard us from the errors and the dangers to which false reasoning exposes us [2]. An approximation of this kind, whenever it is carefully deduced, is always preferable to the most specious reasoning. Let us therefore try to give some general rules for arriving at such an approximation. ^a

The probability of the decisions of an assembly depends on the plurality of the votes, and on the knowledge and impartiality of its members [1]. So many passions and particular interests ^a so ^a often make themselves known there, that it is impossible to calculate this probability. There are, however, some general results, dictated by simple common sense and confirmed by the {probability} calculus. Suppose, for example, that the assembly is ill-informed about a matter submitted for decision, and that this matter requires careful consideration or that the truth in that respect is contrary to received prejudices, so that the odds are more than 1 to 1 that each voter will make a mistake. Then the decision of the majority will probably be wrong, and the larger the assembly, the more ^b well-founded ^b will the fear of this happening be. It is important, then, for the public welfare, that assemblies should have to decide only on matters within the comprehension of the majority: it is also important that information be generally disseminated, and that by good works based on reason and experience, those who are called upon to decide the lot of their fellowmen or to rule them should be enlightened, and should be forewarned against false judgements and the prejudices of ignorance. Scholars have had frequent occasion to remark that first appearances can often be deceptive, and that the truth is not always probable {or plausible}.

Analysis confirms what simple common sense dictates to us, namely, that the more judges there are, and the better informed they are, so much the more probable will it be that the decisions are just [1]. Thus it is important that courts of appeal [2] satisfy these two conditions. The lower courts {i.e. courts of the first instance} [3], attempting to bring those subject to their jurisdiction closer together, offer them the advantage of a first judgment that is already probably just, and with which they are often content, be it in coming to terms, or be it in abandoning their claims. But if the uncertainty of the lawsuit and its importance cause a litigant to have recourse to a court of appeal, he ought to find, in a greater probability of obtaining an equitable judgment, greater protection of his property, and compensation for the trouble and expense that new proceedings entail. This did not happen in the institution of mutual appeal [4] from the courts of the department, an institution which thereby was very prejudicial to the interests of the citizens. It would perhaps be proper and conformable to the probability calculus to require a majority of at least two votes, in a court of appeal, to quash the sentence of the lower court. This result would be obtained if the appeal court were composed of an even number of judges, the sentence being upheld in the case of equality of votes.

There is a very simple way of constructing mortality tables [1]. ^a From the civil registers a large number of individuals whose births and deaths are shown, are taken. Then one determines how many of these individuals have died in the first year of life, how many in the second year, and so on. From this one may deduce how many individuals were alive at the beginning of each year, and one writes this number in the table next to that which indicates the year. Thus one writes the number of births next to 0, the number of children who reached one year next to the year 1, the number of children who reached two years next to year 2, and so on. ^a But as mortality is very high in the first two years of life, it is necessary, for greater accuracy, to indicate in these early years the number of survivors at the end of each half-year.

Let us recall here what has already been said about expectation [1]. We have seen that in order to find the advantage that results from several simple events, some of which produce a gain and others a loss, it is necessary to add the products of the probability of each favourable event and the gain that it procures, and to subtract from ^a their ^a sum that of the products of the probability of each unfavourable event and the attendant loss [2]. But whatever the advantage expressed by the difference between these sums may be, a single event composed of these simple events in no wise guarantees against the fear of experiencing a ^α loss. One might imagine that this fear ought to decrease as the compound event is repeated more and more often. Probabilistic analysis leads to the following general theorem:

By the repetition of an advantageous event, be it simple or compound, the real benefit becomes more and more probable, and it increases continually. It becomes certain under the hypothesis of an infinite number of repetitions; and on dividing it by ^b this ^b number, the quotient, or the mean benefit of each event, is the mathematical expectation itself — or the relative advantage of the event. The same thing holds for a loss that becomes certain in the long run, if the event is in the least disadvantageous.

The mind, like the sense of sight, has its illusions [1]; and just as touch corrects those of the latter, so thought and calculation ^α correct the former. Probability based on daily experience, or exaggerated by fear or hope, affects us more than a larger probability that is only a simple result of calculation. Thus, in return for small gains, we have no fear at all in exposing our lives to risks much less unlikely than the drawing of a quine in the French lottery; and yet one would not choose to get the same benefits, with the certainty of losing one’s life if a quine were to occur [2].

Induction, analogy, hypotheses founded on facts and continually corrected by new observations, a fortunate intuition, given by nature and strengthened by numerous comparisons of the things it suggests with experience — these are the principal means of arriving at the truth [1].

The ratios of favourable to unfavourable chances for players in the simplest games have been known for a long time [1]. The stakes and the bets were settled by these ratios. But before Pascal and Fermat, the principles and the methods for calculating such things were not known, and rather complicated questions of this kind had not been solved. Thus it is to these two great mathematicians that we must attribute the fundamentals of the science of probabilities, the discovery of which may be ranked among the remarkable things that have made the 17th century famous — the century that has brought the greatest credit to the human mind. The main problem that they ^α resolved, by different means, consists, as we have already seen [2], in distributing the stakes fairly between players who are of equal ability and who have agreed to cry quits before the game is finished. The condition of play is that he who first reaches a given number of points ^a (different for each of the players) ^a , will win the game. It is clear that the distribution ought to be made proportionally to the players’ respective probabilities of winning this game, probabilities that depend on the numbers of points each of them still lacks. Pascal’s method is very ingenious; it is basically only ^β the partial difference equations ^b of ^b this problem ^c applied to the determinination of ^c the successive probabilities of the players, working one’s way from the smallest numbers upwards. This method is restricted to the case of two players; whereas Fermat’s method, based on combinations, extends to any number of players whatsoever.