320 INDUCTION. 



overlooked in Laplace's statement, or so vaguely indicated as neither 

 to be suggested to the reader, nor kept in view by the WTiter himself. 

 To a calculation of chances, according to Laplace, two things are 

 necessary: we must know that of several events some one will cer- 

 tainly happen, and no more than one ; and we must not know, nor 

 have any reason to expect, that it will be one of these events 

 rather than another. I contend that these are not the only requis- 

 ites, and that anotber supposition is necessary. This supposition it 

 might be imagined that Laplace intended to indicate, by saying 

 that all the events must be equally possible {egalement possibles). 

 But his next sentence shows that, by this expression, he did not 

 mean to add anything to the two conditions which he had already 

 suggested. " The theory of chances consists in reducing all eventa 

 of the same kind to a certain number of cases equally possible, 

 that is, such that we ax'e equally undecided as to their existence ; and 

 to determine the number of these cases which are favorable to the 

 event of which the probability is sought." By " events equally possi- 

 ble," then, he only means events " such that we are equally undecided 

 as to their existence ;" that we have no reason to expect one rather 

 than another ; which is not a third condition, but the second of the 

 two previously specified. I, therefore, feel warranted in affirming 

 that Laplace has overlooked, in this general theoretical statement, 

 a necessary part of the foundation of the doctrine of chances. 



§ 2. To be able to pronounce two events equally probable, it is not 

 enough that we should know that one or the other must happen, and 

 should have no ground for conjecturing which. Experience must 

 have shown that the two events are of equally frequent occurrence. 

 Why, in tossing up a halfpenny, do we reckon it equally probable 

 that we shall throw cross or pile ] Because experience has shown 

 that in any gi'eat number of throws, cross and pile are throwTi about 

 equally often ; and that the more throws we make, tlie more nearly 

 the equality is perfect. We call the chances even, because if we 

 stake equal sums, and play a certain large number of times, experi- 

 ence proves that our gains and losses will about balance one another ; 

 and will continue to do so, however long afterwards we continue play- 

 ing : while on the contrary, if we give the slightest odds, and play a 

 great number of times, we are sure to lose ; and the longer we con- 

 tinue playing, the greater losers we shall be. If experience did not 

 prove this, we should proceed as much at haphazard in staliing equal 

 sums as in laying odds ; we should have no more reason for expecting 

 not to be losers by the one wager than by the other. 



It would indeed require strong evidence to persuade any rational 

 person that by a system of operations upon numbers, our ignorance 

 can be coined into science ; and it is doubtless this strange pretension 

 which has driven a profound thinker, M. Comte, into the contrary 

 extreme of rejecting altogether a doctrine which, however imperfectly 

 its principles may sometimes have been conceived, receives daily veri- 

 fi.cation from the practice of insurance, and from a great mass of other 

 positive experience. The doctrine itself is, I conceive, soimd, but the 

 manner in which its foundations have been laid by its gi-eat teachers is 

 most seriously objectionable. Conclusions respecting the probability 

 of a fact rest not upon a diflerent, but upon the very same ba^is, as 



