24 On the Probability of Testimony and Arguments, 



fcrcncc of the truth of B from that of A 



and that of the converse 



1-X 



Let us analyze the problem more in detail. Suppose the pro- 

 bability of A as proved directly 



the probability of the inference by which B is deduced from A 



and the probability of the disproof of B = X, where, if x v x 2i 

 &c. be separate arguments against B, 



X=I-(l-#)(l-# 2 ) .... 



Then calling the inference C, we have the following cases : — 

 1. A, B, and C all true; probability wy(l— X). 



2. 



A true, B and C false 



)i 





w(l-y)X. 



3. 



A and B true, C false 



i> 





w(l-y)(l-X). 



4. 



A false, B and C true 



a 





(i-wMi-1). 



5. 



A and B false, C true 



}) 





(1-*%X. 



G. 



A and C false, B true 



)) 





(l-w)(l-y)(l-X). 





7. All three false 





Sum 



(l-i»)(l- V )X. 

 . = 1— wyX. 



the only supposition which is necessarily impossible being, that 

 A and C are true, and B false. Hence on the whole the pro- 

 bability of A 



of the inference C 

 ofB 



1— wz/X ' 

 y(l-«X) , 



1 — wyX ' 

 1-X 



1— wyX. 



The odds in favour of A are w—wyK. to I — w; that is, they are 

 diminished in proportion to the strength of y and X combined ; 

 the odds in favour of B are 1 — X to X — wylL; these, therefore, 

 are increased similarly. We deduce the important consequence 

 that, as regards the evidence of A, it is precisely the same thing 



