150 Mr. TOZER, on THE FORCE OF TESTIMONY 



Take then, 



A, ^2 ^ 



2A' 2A' 2A' 



the respective probabilities of the happening of the events E^ ... E„ as derived from our knowledge 

 of the nature of the events themselves; 



26' 26' 26' 



the probabilities that they will be respectively believed, the witness being deceived ; 



2a 2a 2a 



the probabilities that they will respectively be alleged, the witness not believing the one alleged 

 to have occurred ; 



u the probability that the witness is not deceived ; 



V that the testimony is not given with a knowledge that it is false ; 



Pi that the occurrence of £, would cause the allegation to be made; 



p. that the occurrence of £,• any one of the events E^ ... E„ would do so; 



Tfi that the fact alleged is true. 



Now the occurrence of £, will cause the allegation to be made, if the witness be neither deceived 

 nor intendino- to deceive, of which the probability is uv ; and also, if both the one and the other, 

 provided the event chosen for the purpose of deceiving be that which in fact occurred. 



The probability of the hypothesis is (l - u) (1 - t>), and the probability of the particular mode 

 of being deceived being the believing in the occurrence of some one of the events E,, other than 



£ is '- — , since £, cannot be believed, but if E, be believed, the probability that E^ will 



26 - 6, 



be alleged, is ^ , since E, will not be alleged ; the probability therefore that E, will be 



° 2a - a, 



, , . f^i 6, 



believed, and Ei alleged is -- — r- . = . 



2o — 0, Za — a, 



And the whole probability that E^ will be alleged on the hypothesis is 



«i [v b 6i I 



26 - 6, \ 2a - a 2a - o,J 



Hence, 



ai f 6 ^1 I 



Pi = uv + (1 - u) (1 - v) —- IS.- :=^ } . 



^ 25 - 6, I 2a - a 2a - aj 



Again, the occurrence of £, will cause the allegation to be made, first when the witness is 



deceived, and does not intend to deceive, but believes E-^ to have occurred. 



The probability of the hypothesis is (l - u)v. 



6, 

 that jE, will be believed, and therefore alleged -— — . 



2o - 6; 



Secondly, when not deceived, but intending to deceive. 

 The probability of the hypothesis is u(l - v)\ 



that £, will be alleged ~ . 



' ^ 2a - Oj 



Thirdly, when both deceiving and deceived provided among the modes of deceiving the allega- 

 tion of the occurrence of £, be not cho.sen. 



