434 Prof. Boole on a General Method in 



balanced. Now the event an/ can only happen through the com- 

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

 some one of the b cases in which y happens, while nothing pre- 

 vents us from supposing any one of the m cases in which a: hap- 

 pens or fails from combining with any orie of the n cases in 

 which y happens or fails. There are thus ab cases in which the 

 event xy happens, to mn cases which are either favourable or 

 unfavourable to its occurrence. Nor have we any reason to 

 assign a preference to any one of those cases over any other. 



Wherefore the probability of the event xy is — . Or if we 



represent the probability of the event x by jo, that of the event y 

 by q, the probability of the combination xy is pq. 



It cannot be disputed that the above is a rigorous consequence 

 of the definition adopted. That new information might alter 

 the value of Prob. xy is only in accordance with the principle 

 (already exemplified from Laplace) of the relative character of 

 probability. It is only so far forth as they are known, that the 

 connexions, causal or otherwise, of events can aff'ect expectation. 

 Let it be added, that the particular result to which we have been 

 led is perfectly consistent with the well-known theorem, that if 

 X and y are known to be independent events, the probability of 

 the event xy is pq. The difference between the two cases con- 

 sists not in the numerical value of Prob. xy, but in this, that if 

 we are sure that the events x and y are independent, then are 

 we sure that there exists between them no hidden connexion, the 

 knowledge of which would affect the value of Prob. xy ; whereas 

 if we are not sure of their independence, we are sensible that 

 such connexions may exist. Again, it is perfectly consistent 

 with the known theorem, that if the probability of x is p, and 

 the probability that if x happen y will happen is q, then the 

 probability of the combination xy is pq. For if we know nothing 

 of the connexion ofx and y, the occurrence of x will not affect our 

 expectation of the occurrence of y, so that the probability that if 

 X happen y will happen, will, in the actual state of our informa- 

 tion, be the same as the simple probability of y, i. e. as q. 



4. As from the simple data Prob. x=p, Prob. y=qwe deduce 

 Prob. xy==pq, so from the same data we should have 



?roh.x(l-y)=p{l-q), Vroh.{l-x)(l-y) = {l-p){l-q)kc. 



And generally it may be shown, that if the probabilities of any 

 events x, y, z are simply given, the probability of any combina- 

 tion of them expressed by F {x,y, z . .) will be found by sub- 

 stituting in that expression for x,y,z . . their given probabilities. 

 The general principle involved in the above deductions may be 

 thus stated. 



