632 PROFESSOR BOOLE ON THE COMBINATION 
well as of z. A circumstance is an event—a state of things which comes to pass, 
or has come forth—evenit. 
The data leave wholly arbitrary the probabilities of the event v and y. Thus 
p and q are conditional probabilities; p is the probability that if the event a 
occur, the event will occur; ¢ is the probability that if the event y occur, the 
event 2 will occur. Hence 
_ Prob. xy _ Prob. yz 
Pr ipa ae UT wpapng wpe cals eect or ee 
Our object is to determine the probability that if the events x and y both occur, 
the event 2 will occur. We have therefore to seek the value of the fraction 
Prob. ayz 
Prob. zy 
or, as for our present purpose it is more convenient to say, of 
Prob. xyz 
Prob. wyz+ Prob wyz 
(2) 
In seeking the value of Prob. wyz, which we shall represent by u, the formal 
statement of our data and queesitum will therefore be 
Given { 
Required Prob. wyz. 
Prob. w=c, Prob. y=¢ 
Prob. wz=cp, Prob. yz=c'q (3) 
e and ¢’ being arbitrary constants expressing the unknown probabilities of the 
events w and y. 
A misconception may here arise respecting the meaning of Prob. a, Prob. y, 
which it is worth while to anticipate. In the case of testimony, Prob. x would 
not mean the probability that a testimony would be borne, but the probability 
that the particular kind of testimony actually recorded considered with reference 
to its object, credibility, &c., would be borne. Testimonies differ, not merely as 
to their degree of credibility, but as to their wneapectedness—as to the surprise 
which they occasion. And it is, I think, matter of personal experience that this 
unexpectedness is in itself an element affecting the strength of that expectation 
which combined testimonies produce. So, too, if «and y are facts of observation, 
e.g., observed symptoms of a disease 2, the probability of that disease, when both 
symptoms present themselves, is not determined by the strength of the separate 
presumptions merely, but is consciously increased by our knowledge of the rarity 
of the symptoms themselves. And thus the elements Prob. x Prob. y, which 
have been introduced by a formal necessity of the statement of the problem are 
seen to belong to the very matter of its solution. 
Making 
az=8, y2=t, xyi=,; 
