IN CASES OF LEGAL EVIDENCE. 151 



The probability of the hypothesis is (l - u) (l - ») ; 



that the belief in the occurrence of E. is the mode of bains- deceived, ' 



26 - 6i ' 



that jB] will be alleged, E, being believed, 



that Ey will be believed and £, alleged, 



2a - a, ' 

 a, b. 



26 — 6j 2<i — a, 

 the whole probability, on the hypotheses, that £, will be alleged, is 



^1 {^ __b 6^ bj 1 



> — 6; ( 266 — a 2cs — a, 2a - aj 



26 



£, and Ej being excluded in the several values of £„ because one is alleged and the other happened. 

 Hence, 



26 - o, 2a — a,- 26 - o,. (. 2a -a 2a - Oj 2a — a,-) 



A J 2A 1 



And TTi = 



•Eh ^h p,h, ^' • 



The value of Epih^ being 



2pA- = (1 - «)^'6, (2 ^A^ - — ^1 + «(l - .)a, {2 — ^ - -A_| 

 (.26-6 26 - 6, J t 2a - o 2a - a^) 



+ (l-M)(i-ri)a,[^| ^^_^ 2a-ajr 26-6 26-6j~l (26-6)(2a-a) " (26-6,)(2a-a,)jJ' 



It is here supposed that the occurrence of an event, which is not believed to have occurred, will 

 not affect the disposition to believe in the occurrence of any one event which did not occur in 

 preference to any other ; and that the disposition to allege the occurrence of any event which 

 is not believed to have occurred in preference to any other, is independent of the event which is 

 believed ; if this assumption be not made, the values of a will be different for different values of r, 

 and the values of 6 different for the different values of i, but the process will be the same. 



The expression for 2p;Aj is adapted to the case of all the circumstances by which the belief 

 or veracity of the witness can be influenced being known ; when the data are less complete the 

 expression becomes much simplified, the result of course becoming less accurate, as from the insuf- 

 ficiency of the data it must do. 

 Taking 



r( b b^ \ \ h a^ ^i 1 



+ (,-.)(.- .) L|2^^-^ - ^^-^j |2- ^^-^ - ^^-^j 



h a,.b 



ai6i "1-1 



a,) (26 - 6,)J J 



h, (26 - 6) (2a - a) (2a 



If the data be so incomplete, that among the events by the occurrence of which, either the 

 belief of the witness, or his disposition to allege one fact in preference to another, is influenced, no 



