770 PROBABILITY result of the mathematician will be but an ignoratio elencki 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 pro bability 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 astro nomer 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 oat correct, if we had no other source of informa tion ; 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 approxi mations ; 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 depres sion 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 felloAv-jurors, being left out of the account. Again, the value of an expectation to an indi vidual is taken to be measured by the sum divided by his present fortune, though it is clearly affected by 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~ l , 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, itc., 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 con crete, 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 pre sented 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 pre sent 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, Wh itworth 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, <tc., 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.