k priori Probabilities. 207 



resulted from any other mean. It suffices to infer inductively 

 that, because groups of statistics having that ratio as mean 

 have been in great numbers presented in the past, such groups 

 will be reproduced in the future ; then to take the Mean as 

 the representative of the group ; by the light of common 

 sense, or according to a more formal principle to be set forth 

 hereafter. The case of observations and statistics of different 

 weight presents indeed a difficulty, but not perhaps an insu- 

 perable one, in the way of this procedure. Although the 

 objection seems to have a good deal of force, yet it cannot 

 reasonably be allowed to dispense altogether with the Calculus. 

 At least in cases where our data are not indefinitely numerous 

 we must apply a stroke of inverse probability. And even in 

 cases more favourable to unaided induction, though without the 

 calculus we might be confident that a discrepancy in statistics 

 corresponded to a difference of cause (for example, in the case 

 of male compared with female births), yet could we have an 

 adequate conception of the degree of that assurance without 

 the mathematical calculations of a Laplace ? In fine, is any 

 one competent to assert that those reasonings of the greatest 

 intellects * should be put into the fire ? 



II. If, then, a priori probabilities are required, it behoves 

 us to consider how far that requirement is fulfilled. 



(1) It is fulfilled sufficiently to allow of a mathematical, 

 though not a numerical, inference in cases where the a posteriori 

 probability has a limiting value, provided that an involved 

 a priori probability is not extreme. For example, in the first 

 problem of our first paragraph, when n is indefinitely large, 



we may neglect the correction introduced, provided that 



tf(«) 



is finite ; which seems to be in general reasonably certain. 

 And similarly in the second problem, and subject to a similar 

 proviso, the correction in comparison with the term corrected 

 is neglectible when n is indefinitely large. The remark ex- 

 tends to the problem intermediate between III. and IV. It 

 extends also to the important theorem which assigns the 

 probability that a difference in observations or statistics cor- 

 responds to a difference in cause. This is well indicated by 

 Cournot (with regard to statistics at least), though he hardly 

 seems aware that his formula conr ecting the extent of error 

 and the probability of its occurrence! has no numerical validity 

 upon his view of a priori probabilities. Again, the indeter- 



* See e. y. Laplace, Essai Philosophique, p. lix, 3rd edit., Theorie Ana- 

 lytique, Book II. p. 31 ; Herschel, ' Essays,' p. 422 (Review of Quetelet) ; 

 De Morgan, Encycl. Metropol. §§ 145-6. 



t Exposition de la Theorie des Chances, ch. viii. 



