The mathematical theory of probability is a science which aims at reducing to calculation, where possible, the amount of credence due to propositions or statements, or to the occurrence of events, future or past, more especially as contingent or dependent upon other propositions or events the probability of which is known.
Any statement or (supposed) fact commands a certain amount of credence, varying from zero, which means conviction of its falsity, to absolute certainty, denoted by unity. An even chance, or the probability of an event which is as likely as not to happen, is represented by the fraction 1/2. It is to be observed that 1/2 will be the probability of an event about which we have no knowledge whatever, because if we can see that it is more likely to happen than not, or less likely than not, we must be in possession of some information respecting it. It has been proposed to form a sort of thermometrical scale, to which to refer the strength of the conviction we have in any given case. Thus if the twenty-six letters of the alphabet have been shaken together in a bag, and one letter be drawn, we feel a very feeble expectation that A has been the one taken. If two letters be drawn, we have still very little confidence that A is one of them; if three be drawn, it is somewhat stronger; and so on, till at last, if twenty-six be drawn, we are certain of the event, that is, of A having been taken.
Probability, which necessarily implies uncertainty, is a consequence of our ignorance. To an omniscient Being there can be none. Why, for instance, if we throw up a shilling, are we uncertain whether it will turn up head or tail? Because the shilling passes, in the interval, through a series of states which our knowledge is unable to predict or to follow. If we knew the exact position and state of motion of the coin as it leaves our hand, the exact value of the final impulse it receives, the laws of its motion as affected by the resistance of the air and gravity, and finally the nature of the ground at the exact spot where it falls, and the laws regulating the collision between the two substances, we could predict as certainly the result of the toss as we can which letter of the alphabet will be drawn after twenty-five have been taken and examined.
The probability, or amount of conviction accorded to any fact or statement, is thus essentially subjective, and varies with the degree of knowledge of the mind to which the fact is presented (it is often indeed also influenced by passion and prejudice, which act powerfully in warping the judgment), -- so that, as Laplace observes, it is affected partly by our ignorance partly by our knowledge. Thus, if the question were put, Is lead heavier than silver? some persons would think it is, but would not be surprised if they were wrong; others would say it is lighter; while to a worker in metals probability would be superseded by certainty. Again, to take Laplace's illustration, there are three urns A, B, C, one of which contains black balls, the other two white balls; a ball is drawn from the urn C, and we want to know the probability that it shall be black. If we do not know which of the urns contains the black balls, there is only one favourable chance out of three, and the probability is said to be 1/3. But if a person knows that the urn A contains white balls, to him the uncertainty is confined to the urns B and C, and therefore the probability of the same event is 1/2 . Finally to one who had found that A and B both contained white balls, the probability is converted into certainty.
In common language, an event is usually said to be likely or probable if it is more likely to happen than not, or when, in mathematical language, its probability exceeds 1/2; and it is said to be improbable or unlikely when its probability is less than 1/2. Not that this sense is always adhered to; for, in such a phrase as "It is likely to thunder to-day," we do not mean that is more likely than not, but that in our opinion the chance of thunder is greater than usual; again, "Such a horse is likely to win the Derby," simply means that he has the best chance, though according to the betting that chance may be only 1/6. Such unsteady and elliptical employment of words has of course to be abandoned and replaced by strict definition, at least mentally, when they are made the subjects of mathematical analysis. Certainty, or absolute conviction, also, as generally understood, is different from the mathematical sense of the word certainty. It is very difficult and often impossible, as is pointed out in the celebrated Grammar of Assent, to draw out the grounds on which the human mind in each case yields that conviction, or assent, which, according to Newman, admits of no degrees, and either is entire or is not at all.  If, when walking on the beach, we find the letters "Constantinople" traced on the sand, we should feel, not a strong impression, but absolute certainty, that they were characters not drawn at random, but by one acquainted with the word so spelt. Again, we are certain of our own death as a future event; we are certain, too, that Great Britain is an island; yet in all such cases it would be very difficult, even for a practised intellect, to present in logical form the evidence, which nevertheless has compelled the mind in each instance to concede the point.  Mathematical certainty, which means that the contrary proposition is inconceivable, is thus different, though not perhaps as regards the force of the mental conviction, from moral or practical certainty. It is questionable whether the former kind of certainty is not entirely hypothetical, and whether it is ever attainable in any of the affairs or events of the real world around us. The truth of no conclusion can rise above that of the premises, of no theorem above that of the data. That two and two make four is an incontrovertible truth; but before applying even it to a concrete instance we have to be assured that there were really two in each constituent group; and we can hardly have mathematical certainty of this, as the strange freaks of memory, the tricks of conjurors, &c. have often made apparent.
There is no inore remarkable feature in the mathematical theory of probability than the manner in which it has been found to harmonize with, and justify, the conclusions to which mankind have been led, not by reasoning, but by instinct and experience, both of the individual and of the race. At the same time it has corrected, extended, and invested them with a definiteness and precision of which these crude, though sound, appreciations of common sense were till then devoid. Even in cases where the theoretical result appears to differ from the common-sense view, it often happens that the latter may, though perhaps unknown to the mind itself, have taken account of circumstances in the case omitted in the data of the
theoretical problem. Thus, it may be that a person accords a lower degree of credence to a fact attested by two or more independent witnesses than theory warrants, -- the reason being that he has unconsciously recognized the possibility of collusion, which had not been presented among the data. Again, it appears from the rules for the credibility of testimony that the probability of a fact may be diminished by being attested by a new witness, viz., in the case where his credibility is less than 1/2. This is certainly at variance with our natural impression, which is that our previous conviction of any fact is clearly not weakened, however little it be intensified, by any fresh evidence, however suspicious, as to its truth. But on reflexion we see that it is a practical absurdity to suppose the credibility of any witness less than 1/2 --that is, that he speaks falsehood oftener than truth -- for all men tell the truth probably nine times out of ten, and only deviate from it when their passions or interests are concerned. Even where his interests are at stake, no man has any preference for a lie, as such, above the truth; so that his testimony to a fact will at worst leave the antecedent probability exactly what it was.
A celebrated instance of the confirmation and completion by theory of the ordinary view is afforded by what is known as James Bernoulli's theorem. If we know the odds in favour of an event to be three to two, as for instance that of drawing a white ball from a bag containing three white and two black, we should certainly judge that if we make five trials we are more likely to draw white three times and black twice than any other combination. Still, however, we should feel that this was very uncertain; instead of three white, we might draw white 0, 1, 2, 4, or 5 times. But if we make say one thousand trials, we should feel confident that, although the numbers of white and black might not be in the proportion of three to two, they would be very nearly in that proportion. And the more the trials are multiplied the more closely would this proportion be found to obtain. This is the principle upon which we are continually judging of the possibility of events from what is observed in a certain number of cases.  Thus if, out of ten particular infants, six are found to live to the age of twenty, we judge, but with a very low amount of conviction, that nearly six-tenths of the whole number born live to twenty. 3ut if, out of 1,000,000 cases, we find that 600,000 live to be twenty, we should feel certain that the same proportion would be found to hold almost exactly were it possible to test the whole number of cases, say in England during the 19th century. In fact we may say, considering how seldom we know a priori the probability of any event, that the knowledge we have of such probability in any case is entirely derived from this principle, viz., that the proportion which holds in a large number of trials will be found to told in the total number, even when this may be infinite, -- the deviation or error being less and less as the trials are multiplied.
Such no doubt is the verdict of the common sense of mankind, and it is not easy to say upon what considerations it is based, if it be not the effect of the unconscious habit which all men acquire of weighing chances and probabilities, in the state of ignorance and uncertainty which human life is. It is now extremely interesting to see the results of the unerring methods of mathematical analysis when applied to the same problem. It is a very difficult one, and James Bernoulli tells us he reflected upon it for twenty years. His methods, extended by De Moivre and Laplace, fully confirm the conclusions of rough common sense; but they have done much more. They enable us to estimate exactly how far we can rely on the proportion of cases in a large number of trials, truly representing the proportion out of the total number -- that is, the real probability of the event. Thus he proves that if, as in the case above mentioned, the real probability of an event is 3/5, the odds are 1000 to 1 that, in 25,550 trials, the event shall occur not more than 15,841 times and not less than 14,819 times, -- that is, that the deviation from 15,330, or 3/5 of the whole, shall not exceed 1/50 of the whole number of trials.
The history of the theory of probability, from the celebrated question as to the equitable division of the stakes between two players on their game being interrupted, proposed to Pascal by the Chevalier de Méré in 1654, embracing, as it does, contributions from almost all the great names of Europe during the period, down to Laplace and Poisson, is elaborately and admirably given by Mr Todhunter in his History of the subject, now a classical work. It was not indeed to be anticipated that a new science which took its rise in games of chance, and which had long to encounter an obloquy, hardly yet extinct, due to the prevailing idea that its only end was to facilitate and encourage the calculations of gamblers, could ever have attained its present status -- that its aid should be called for in every department of natural science, both to assist in discovery, which it has repeatedly done (even in pure mathematics), to minimize the unavoidable errors of observation, and to detect the presence of causes as revealed by observed events. Nor are commercial and other practical interests of life less indebted to it :  wherever the future has to be forecasted, risk to be provided against, or the true lessons to be deduced from statistics, it corrects for us the rough conjectures of common sense, and decides which course is really, according to the lights of which we are in possession, the wisest for us to pursue. It is sui generis and unique as an application of mathematics, the only one, apparently, lying quite outside the field of physical science. De Moivre has remarked that, "some of the problems about chance having a great appearance of simplicity, the mind is easily drawn into a belief that their solution may be attained by the mere strength of natural good sense "; and it is with surprise we find that they involve in many cases the most subtle and difficult mathematical questions. It has been 'ound to tax to the utmost the resources of analysis and the powers of invention of those who have had to deal vith the new cases and combinations which it has presented. Great, however, as are the strictly mathematical difficulties, they cannot be said to be the principal. Especially in the practical applications, to detach the problem from its surroundings in rerum natura, discarding what is non-essential, rightly to estimate the extent of our knowledge respecting it, neither tacitly assuming as known what is not known, nor tacitly overlooking some datum, perhaps from its very obviousness, to make sure that events we are taking as independent are not really connected, or probably so, -- such are the preliminaries necessary before the question is put in the scientific form to which calculation can be applied, and failing which the result of the mathematician will be put an ignoratio elenchi -- a correct answer, but to a different question.
From its earliest beginnings, a notable feature in our subject has been the strange and insidious manner in which errors creep in -- often misleading the most acute minds, as in the case of D'Alembert -- and the difficulty of detecting them, even when one is assured of their presence by the evident incorrectness of the result. This is probably in many cases occasioned by the poverty of language obliging us to use one term in the same context for different things -- thus introducing the fallacy of ambiguous middle; e.g., the same word "probability" referring to the same event may sometimes mean its probability before a certain occurrence, sometimes after; thus the chance of a horse winning the Derby is different after the Two Thousand from what it was before. Again, it may mean the probability of the event according to one source of information, as distinguished from its probability taking everything into account; for instance, an astronomer thinks he can notice in a newly-discovered planet a rotation from east to west; the probability that this is the case is of course that of his observations in like cases turning out correct, if we had no other source of information; but the actual probability is less, because we know that at least the vast majority of the planets and satellites revolve from west to east. It is easy to see that such employment of terms in the same context must prove a fruitful source of fallacies; and yet, without wearisome repetitions, it cannot always be avoided. But, apart from mere logical errors, the main stumbling-block is no doubt the uncertainty as to the limits of our knowledge in each case, or -- though this may seem a contradiction in terms -- the difficulty of knowing what we do know; and we certainly err as often in forgetting or ignoring what we do know, as in assuming what we do not. It is a not uncommon popular delusion to suppose that if a coin has turned up head, say five times running, or the red has won five times at roulette, the same event is likely to occur a sixth time; and it arises from overlooking (perhaps from the imagination being struck by the singularity of the occurrence) the a priori knowledge we possess, that the chance at any trial is an even one (supposing all perfectly fair); the mind thus unconsciously regards the event simply as one that has recurred five times, and therefore judges, correctly, that it is very likely to occur once more. Thus if we are given a bag containing a number of balls, and we proceed to draw them one by one, and the first five drawn are white, the odds are 6 to 1 that the next will be white, -- the slight information afforded by the five trials being thus of great importance, and strongly influencing the probabilities of the future, when it is all we have to guide us, but absolutely valueless, and without influence on the future, when we have a priori certain information. The lightest air will move a ship which is adrift, but has simply no effect on one securely moored.
It is not to be supposed that the results arrived at when the calculus of probabilities is applied to most practical questions are anything more than approximations; but the same may be said of almost all such applications of abstract science. Partly from ignorance of the real state of the case, partly from the extreme intricacy of the calculations requisite if all the conditions which we do or might know are introduced, we are obliged to substitute in fact, for the actual problem, a simpler one approximately representing it. Thus, in mechanical questions, assumptions such as that the centre of gravity of an actual sphere is at its centre, that the friction of the rails on a railway is constant at different spots or at different times, or that in the rolling of a heavy body no depression Is produced by its weight in the supporting substance, are instances of the convenient fictions which simplify the real question, while they prevent us accepting the result as more than something near the truth. So in probability, the chance of life of an individual is taken from the general tables (unless reasons to the contrary are very palpable) although, if his past history, his mode of life, the longevity of his family, &c., were duly weighed, the general value ought to be modified in his case; again, in attempting to estimate the value of the verdict of a jury, whether unanimous or by a majority, each man is supposed to give his honest opinion, -- feeling and prejudice, or pressure from his fellow-jurors, being left out of the account. Again, the value of an expectation to an individual is taken to be measured by the sum divided by his present fortune, though it is clearly affected hy other circumstances, as the number of his family, the nature of his business, &c. An event has been found to occur on an average once a year during a long period: it is not difficult to show that the chance of its happening in a particular year is 1 - e-1, or 2 to 1 nearly. But, on examining the record, we observe it has never failed to occur during three years running. This fact increases the above chance ; but to introduce it into the calculation at once renders the question a very difficult one. Even in games of chance we are obliged to judge of the relative skill of two players by the result of a few games; now one may not have been in his usual health, &c., or may have designedly not played his best; when he did win he may have done so by superior play, or rather by good luck; again, even in so simple a case as pitch and toss, the coin may, in the concrete, not be quite symmetrical, and the odds of head or tail not quite even.
Not much has been added to our subject since the close of Laplace's career. The history of science records more than one parallel to this abatement of activity. When such a genius has departed, the field of his labours seems exhausted for the time, and little left to be gleaned by his successors. It is to be regretted that so little remains to us of the inner working of such gifted minds, and of the clue by which each of their discoveries was reached. The didactic and synthetic form in which these are presented to the world retains but faint traces of the skilful inductions, the keen and delicate perception of fitness and analogy, and the power of imagination -- though such a term may possibly excite a smile when applied to such dry subjects -- which have doubtless guided such a master as Laplace or Newton in shaping out each great design -- only the minor details of which have remained over, to be supplied by the less cunning hand of commentator and disciple.
We proceed to enumerate the principal divisions of the theory of probability and its applications. Under each we will endeavour to give at least one or two of the more remarkable and suggestive questions which belong to it, -- especially such as admit of simplification or improvement in the received solutions; in such an article as the present we, are debarred from attempting even an outline of the whole. We will suppose the general fundamental principles to be already known to the reader, as they are to be now found in several elementary works, such as Todhunter's Algebra, Whitworth's Choice and Chance, &c.
Many of the most important results are given under the apparently trifling form of the chances in drawing balls from an urn, &c., or seem to relate to games of chance, as dice, or cards, but are in reality of far wider application, -- this form being adopted as the most definite and lucid manner of presenting the chances of events occurring under circumstances which may be assimilated, more or less closely, to such cases.
768-1 "There is a sort of a leap which most men make from a high probability to absolute assurance ... analogous to the sudden consilience, or springing into one, of the the two images see by binocular vision, when gradually brought within a certain proximity." -- Sir J. Herschel, in Edin. Review, July 1850.
768-2 Archbishop Whately's jeu d'esprit, Historic Doubts respecting Napoleon Bonaparte, is a good example of the difficulties there may be in proving a conclusion the certainty of which is absolute.
769-1 So it said, "the tree is known by its fruits"; "practice is better than theory"; and the universal sense of mankind judges that the safest test of any new invention, system, or institution is to see how it works. So little are we able by a priori speculations to forecast the thousand obstacles and disturbing influences which manifest themselves when any new cause or agent is introduced as a factor in the world's affairs.
769-2 Men were surprised to hear that not only births, deaths, and marriages, but the decisions of tribunals, the results of popular elections, the influence of punishments in checking crime, the comparative values of medical remedies, the probable limits of error in every department of physical inquiry, the detection of causes, physical, social, and moral, nay, even the weight of the evidence and the validity of logical argument, might come to be surveyed with the lynx-eyed scrutiny of dispassionate analysis. -- Sir J. Herschel.