154 Mb. TOZER, on THE FORCE OF TESTIMONY 



■7r\ir..tr-i + A^iTTiTTj (I-ttj) + AaTTj (I-tt^) tts + ZCs (I-tti) TTiTTsirs 



+ fr, (l-TrO'TaTTs + h-^i (1-Ti) (l-7r2)7r3 + fejin (l -7r,) ir^ (l -ttj) + fc, iTi (l -Tr,) (l — ttj) 

 + fe, (l-7r,)'ir2(l— TTa) + k^ (l -tti^ (l -^2) T3 + AiolTj (l — tt,) (1 — TT^) (l -Trj). 



Both the facts beincr certain when all the circumstances concur, the factor of the first term 

 is 1 ; A; , As, /ts are the probabilities that the inferences are true when three only out of the 

 four elementary facts concur, and the remaining one is false; fej, k^, k^, k., when two are true 

 and two false; and kg, k^, k,„ when one only is true; k,, &c. are not or not necessarily the same 

 in the value of P, as in that of P,. 



The evidence in this case is so far complete, and would or would not warrant the conviction 

 of the accused, as P„ did or did not exceed or equal the standard of conviction ; there was 

 however in the particular case a subsequent chain of facts spoken to. 



First, a witness alleged that he sold the accused As after the administering first spoken to: 

 if p, be the probability that this allegation is true, p, will be determined by the formula for tt,. 



Let Q, be the probability of the possession of As by the accused, with knowledge, after this 

 allegation. 



Then Q,= l -(1 - P.) (1 - pO- 



With regard to the administering subsequent to the second purchase, three witnesses severally 

 alleged, that they saw the accused administer a white powder, whose appearance, from their de- 

 scription, corresponded with that of As. 



For the probability that this fact is true, let 9,, r,, «, be the respective values of p in 

 the formula for tti for each of these witnesses respectively, and let p.^ be the probability sought. 

 Then, preserving the remainder of the notation, 



P'^ — '~T~^h 



1 + 2- 9,- )•,■«,.. 



g, r, «i A, 



But the fact of possession, with knowledge, of which the probability is Q,, concurring witli the 

 admitted cause of death, affords, independently of the last fact, some probability of the second 

 administering. 



Hence, if Q., be this latter probability, 



/, , /j, and 4 heing the respective probabilities that As was administered, when it was possessed; 

 and something like it was administered, when it was simply possessed, and simply when some- 

 thing like it was administered ; the cause of the death being in all the cases assumed. 

 After this second series of facts, we get 



P^ = 1 - (1 - P, P,) (1 - Q, QJ = P,P, + Q, Q, (1 - P, P,). 



The numerical values below are assigned for the purpose of completing the illustration, and not 

 with a view to obtain the actual numerical result in the particular case, the assigning of those values 

 is no part of the scientific process, but is determined by a consideration of the situation and character 

 of the witnesses, and of the manner in which they give their testimony. 



The operation is also completed for the purpose of shewing, by the attempt to assign numerical 

 values, that the practical approximation to a correct result must necessarily be a rude one. Though the 

 elementary probabilities are expressed by low numbers, the resulting numbers rapidly become very 

 large ; and to assign at once the value of the resulting probability, without the assistance of the 

 processes of calculation, would be necessarily to assign them very inaccurately; and the process of at 

 once determining the consequences of that value must be affected with at least as great a chance of 

 error. We may, perhaps, in criminal cases, make as small as we please the chance of an innocent 



