It thus appears that the theory of probabilities is simply the re duction to a numerical estimation, in cases which are perfectly known as to the number of events which may happen, of our comparative right to expect one or another event in preference to the rest. In the events of common life, we make estimations of this comparative right, but not numerically, because we are not in sufficient possession of the events which might have happened instead of the one which does happen. In such terms as barely possible, very unlikely, improbable, not improbable, as likely as not, rather likely, highly probable, almost certain, &c., we see a gradation which amounts to a rough attempt to make those comparisons which might be made numerically if the proper data could be obtained. The truth is, that every one naturally admits and practises the fundamental principles of this theory, though often, it be, unskilfully. But it is not to be that persons in general make an investigation like the pre ceding, nor indeed any investigation at all: how is it theft that not only are these principles acquired, however imperfectly, by the majority of mankind, but the mathematical results which are obtained by those who professedly study the subject, are received almost univer sally ; not only aro they felt to be agreeable to common sense in a great majority of cases, but they are soon admitted to be sufficient indications that common sense is wrong, in the few cases in which they at first appear repugnant to it/ The answer to this question leads to another view of the subject.
We all find, by every-day observation, that whenever an event of one kind happens permanently more often than one of another kind, there exists some reason for such frequency of occurrence, which, had it been inquired into before any event happened, would have enabled us to predict the frequency in question. So much is this the case, that if we were to take an observer to an urn in which were black and white balls, but how many of each he is not told, and were to make 1000 draw. big!, replacing the ball drawn after each drawing, and shaking the urn before every trial; if of the thousand drawings 822 were white and 178 black, he would be irresistibly led to conclude that there must be more white balls than black ones in tho urn. Not that this is abso
lutely necessary ; for It is barely possible that there may be only one white ball and 999 black ones, and that by a remarkable coincidence the sole white ball may have come up 822 times out of the 1000 trials. More than this, a person used to observation would conclude, not only that there are more white balls In the urn, but that the proportion of white and black balls does not differ very greatly from that of 322 to 178. There is a disposition, derived from experience, to think that events happen in the long run in some sort of manner connected with the facilities afforded for their happening beforehand : and hence follows a disposition to judge what those facilities were from observed events, as well as to predict events from observation of those facilities; and It may happen that one person would draw his notion of likelihood from the first, and another from the second. Thus, it we were to put the question, " What do you mean by saying that it is 5 to 3 for A against B Y " one person might reply, " I mean that in the long run A will happen 5 times for every 3 times which s happens;" while another might say, "I mean that precedent circumstances are so arranged that for every three contingencies under which B may happen, there are 5 under which A may happen." If these two persons were to dispute which was the true mode of viewing the subject, they would be fighting about nothing ; for both are true, and each of them follows from the other : but if they were to differ as to which is the common mode, we should feel rather disposed to side with the one who was for the first answer.
The following problems, illustrative of the connection between the two modes, are the results of mathematical investigation. The table used is that given in MEAN : hut two decimal places must he made in column A, and four in column B; thus 451 opposite to 4 must be read as .0451 opposite to .04, similarly comes opposite to and '9825 opposite to Pa