Prof. Boole on the Theory of Probabilities. 99 



2nd. That upon the supposition of the absence of such cause, 

 or simple combination of causes, certain results- appearing to us 

 equally probable, the probability of that definite combination of 

 those results which constitutes the effect observed may be defi- 

 nitely calculated. 



3rd. That if the value thus obtained be expressed by p, then 

 the formula (1.) will represent the probability of the existence of 

 such predominant cause or combination of causes. That in that 

 formula we may, following Mr. De Morgan, justly assume c=l, 

 but that there appear to be no grounds further than the analogy 

 of Nature for determining a. [The difficulty here is not that 

 we are choosing among causes equally probable, but that we are 

 attempting to assign the a priori probability of the existence of a 

 condition of things, or in other words, to compare the probabi- 

 lities of its presence and its absence. Now this is a question, 

 the conjectural solution of which will vary with our varying 

 knowledge of the constitution of Nature. Unless, however, we 

 have reason to suppose that the value in question is very small, 

 the general formula will still be available for our general guidance, 

 if not for definite numerical evaluation.] 



Quitting this problem, I shall now notice two others, of which 

 solutions have been given, that appear to me to be defective in 

 generality from the same cause, viz. the non-recognition of the 

 requisite arbitrary constants. 



1st. Given j9 the probability of an event X,andg' the probability 

 of the joint concurrence of the events X and Y : required the 

 probability of the event Y. 



The solution of this problem afforded by the general method 

 described in my last letter is 



Prob. ofY = 9 + c(l-jo), 



where c represents the unknown probability, that if the event X 

 does not take place the event Y will take place. Hence it ap- 

 pears that the limiting probabilities of the event Y are q and 

 1 + g— JO. The result is easily verified. 



The only published solution of this problem with which I am 

 acquainted is 



Prob. ofY=i, 

 P 

 a result which involves the supposition that the events X and Y 

 are independent. This supposition is, however, only legitimate 

 when the distinct probabilities of X and Y are afforded in the 

 data of the question. 



Given the probabilities p and q of the two premises of the syl- 

 logism, 



All Ys are Xs 

 AllZsareYs. 



