OF TESTIMONIES OR JUDGMENTS. 601 



Let a be the number of distinct cases favourable to the event x, out of m dis- 

 tinct cases equally possible, from the comparison of which the probability p has 

 been assigned to the event x. In like manner let b be the number of distinct 

 cases favourable to the event ?/, out of n distinct cases equally possible, from the 

 comparison of which the probability q has been assigned to the event y. Then, 



a -, b 



— =p and -=q. 



m n 



Now the conjunction xy can only come to pass through the combination of 

 some one of the a cases in which x happens, with some one of the b cases in which 

 y happens, at the same time that we have an equal right to suppose that any one 

 of the m cases in which x happens or fails may combine with any one of the n 

 cases in which y happens or fails. To none of these combinations is the mind 

 entitled to attach any preference over any other, Hence there exist ab distinct 

 cases favourable to the conjunction of x and y out of a total of mn distinct and 

 equally possible cases. Thus, by the definition, the probability of the conjunc- 

 tion of x and y will be represented by the product — or pq. 



Here the question may be asked, — Does, then, no difference exist between the 

 case in which the events x and y are known to be independent, and that in which 

 we are simply ignorant of the existence of any connection between them ? I 

 reply that there is none, so far as the numerical estimation of probability is con- 

 cerned. There is, however, an important difference as respects the practical 

 value of the numerical result. If the events x and y are known to be inde- 

 pendent, and to have probabilities p and q, we know that, in the long run, the 

 conjunction xy will tend to recur with a frequency which will be proportional 

 to the magnitude of the fraction pq, We do not know that this will be the case 

 if we are simply ignorant of any connection between x and y. This is the differ- 

 ence referred to, and it is an important one. But it does not affect the calcula- 

 tion of probability as flowing from the definition of its numerical measure. 



8. As from the data Prob. x=p, Prob. y=q,we deduce Prob. xy—pq, so from the 

 same data we should have, adopting the language of the calculus of Logic, 



Prob. x (1-?/) =p{l-q) Prob. (1-x) (l-y) = (l-p) (1-q), 



and so on. Here %(l—y) denotes the compound event which consists in the oc- 

 currence of x conjointly with the non-occurrence of y; (\- x (\-y), the compound 

 event which consists in the joint non-occurrence of both x and y. 

 Extending this mode of investigation, we arrive at the theorem 



Prob. (p(x,y,z..) = (p(p,q,r..). . . . (I) 



where x, y, z, &c, denote any simple events whose probabilities (our only data) 

 arep, q, r . ., and (x, y, z . .) denotes any event which can be expressed by 



VOL. XXI. PART. IV. 7 Z 



