OF TESTIMONIES OR JUDGMENTS. 601 
Let a be the number of distinct cases favourable to the event z, out of m dis- 
tinct cases equally possible, from the comparison of which the probability p has 
been assigned to the event 2. In like manner let b be the number of distinct 
cases favourable to the event y, out of m distinct cases equally possible, from the 
comparison of which the probability g has been assigned to the event y. Then, 
b 
=P, and wat 
Now the conjunction zy can only come to pass through the combination of 
some one of the @ cases in which w happens, with some one of the @ 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 ad distinct 
cases favourable to the conjunction of z and y out of a total of mn distinct and 
equally possible cases. Thus, by the definition, the probability of the conjunc- 
tion of # 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 py and g, we know that, in the long run, the 
conjunction zy will tend to recur with a frequency which will be proportional 
to the magnitude of the fraction yg, 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=g, we deduce Prob. #y=pq, so from the 
same data we should have, adopting the language of the calculus of Logic, 
Prob. x(1—y)=p(1—g) Prob. (l—2) (l—y)=(1—p) (1-9), 
and soon. Here #(1—y) denotes the compound event which consists in the oc- 
currence of # conjointly with the non-occurrence of y; (1—x (1—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. O(a, y,2.-)= O(p,q,7-.). : : 5 (1) 
where «, y, z, &c., denote any simple events whose probabilities (our only data) 
are p, q,7.., and > (a, y, z..) denotes any event which can be expressed by 
VOL. XXI. PART. IV. Giz 
