OF THE CALCULATION OF CHANCES. 379 



CHAPTER XVIII. 



OP THE CALCULATION OF CHANCES. 



§ 1. "Probability," says Laplace,* "has reference partly to our igno- 

 rance, partly to our knowledge. We know that among three or more 

 events, one, and only one, must happen ; but there is nothing leading us to 

 believe that any one of them will happen rather than the others. In this 

 state of indecision, it is impossible for us to pronounce with certainty on 

 their occurrence. It is, however, probable that any one of these events, 

 selected at pleasure, will not take place ; because we perceive several cases, 

 all equally possible, which exclude its occurrence, and only one which fa- 

 vors it. 



" The theory of chances consists in reducing all events of the same kind 

 to a certain number of cases equally possible, that is, such that we are 

 equally undecided as to their existence ; and in determining the number of 

 these cases which are favorable to the event of which the probability is 

 sought. The ratio of that number to the number of all the possible cases 

 is the measure of the probability ; which is thus a fraction, having for its 

 numerator the number of cases favorable to the event, and for its denom- 

 inator the number of all the cases which are possible." 



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

 necessary ; we must know that of several events some one will certainly 

 happen, and no more than one ; and we must not know, nor have any rea- 

 son to expect, that it will be one of these events rather than another. It 

 has been contended that these are not the only requisites, and that Laplace 

 has overlooked, in the general theoretical statement, a necessary part of the 

 foundation of the doctrine of chances. To be able (it has been said) 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 grounds for 

 conjecturing which. Experience must have shown that the two events are 

 of equally frequent occurrence. Why, in tossing up a half-penny, do we reck- 

 on it equally probable that we shall throw cross or pile ? Because we know 

 that in any great number of throws, cross and pile are thrown about equally 

 often ; and that the more throws we make, the more nearly the equality is 

 perfect. We may know this if we please by actual experiment, or by the 

 daily experience which life affords of events of the same general character, 

 or, deductively, from the effect of mechanical laws on a symmetrical body 

 acted upon by forces vai'ying indefinitely in quantity and direction. We 

 may know it, in short, either by specific experience, or on the evidence of 

 our general knowledge of nature. But, in one way or the other, we must 

 know it, to justify us in calling the two events equally probable; and if we 

 knew it not, we should proceed as much at hap-hazard in staking equal sums 

 on the result, as in laying odds. 



This view of the subject was taken in the first edition of the present 

 work ; but I have since become convinced that the theory of chances, as 



* Essai Philosophique sur les Probabilites, fifth Paris edition, p. 7. 



