the Theory of Probabilities. 363 



teriori probability of the existence of C is 



pa + {l-p)b ,-p. 



p{a + u) + {l-p){b + ^) 



Let us suppose that a, b, a, /3 are definitively known. Then 

 i£p were definitively known, this expression would be the com- 

 plete solution of the problem. But if p be not definitively 

 knowTi, it only informs us what would be the solution, supposing 

 any proposed value of ^j were the true one. 



17. Now if we have obtained, no matter how*, so much know- 

 ledge oip, that (j){p)dp expresses the quantity of om- belief that 

 its definitive value is between ;j and p + dp, then (14.) our "pro- 

 visional " value ofp hf^p(f>{p)dp, which we will denote by -ar. 

 Hence, writing the expression (P), as we obviously may, in the 



form ^ (where k, I, m are given quantities), we may substi- 



p-\-)ii 

 tute the provisional value •a for p, and obtain 



kts + l , /J V 



ts + m 



as the solution of the problem. 



But, on the other hand, we may reason as follows : — The 

 product 



i>{p)—, — f^p 



p + m 

 expresses the quantity of oui- belief that the value of jt> is between 

 p andp + dp, and that C was the cause of the event E. Hence 



\^p)^I±ldp (XL) 



p + m 



expresses our whole belief in the existence of C, and is the solu- 

 tion of the problem. 



It is by no means obvious at first sight which of these expres- 

 sions (I.), (II.) ought to be preferred, or what is the difference 

 of jjrinciple involved in the processes by which they were ob- 

 tained. But a little consideration will show that (II.) expresses 

 a real protmona/ sulution ; that is, it expresses our belief in the 

 existence of C, injluencedby the consideration that ire do not possess 

 a dejinitive knowtedye of j). Whereas (I.) expresses a solution 

 obtained by treating a provisional value of p as if it were definitive ; 

 or it is what would be the definitive solution of the problem to 



* The reader will not suppose tli:it I omit any discussion of the jiossibi- 

 lity or mode of assif^ning tlie form of the function (j) because I consider it 

 a subject not requiring (liseussion, but it would be irrelevant to my imme- 

 diate ))ur])()sc. 



/ 



