Testimony and Arguments, 21 



proportion between extraordinary and ordinary events in general. 

 It maybe stated somewhat differently thus: — A witness whose 

 credibility is a, announces an event, the antecedent probability 

 of which is p, and the chance of its being announced without 

 reason r j then the resultant probability is 



#/?(!-— /*) 

 ap{l — r)-f (1 — «)(!— p)r 

 If complete generality is desired, we must introduce separately 

 the chances of the witness being mistaken h 3 and of his intending 

 to deceive/, and also the chances of the event being invented i, 

 or being erroneously supposed to have happened s. It may be 

 worth while to exhibit the formula in this shape. It is obvious 

 that when the event has actually happened, it may or may not 

 be that the causes which would lead to its being either invented 

 or falsely believed have also existed, i. e. pi and ps are possible 

 cases. This consideration simplifies the formula, which will be 

 found to give for the resulting probability of the event announced. 



p {(!-/)(! -k) +fki\ 



P{{l-f){\-k)+fki\ + (\-p)^\-f)ks+fi) ' 



When r—pj the probability just given is reduced to a, and 

 accordingly it frequently happens that, although the event 

 announced is extraordinary, the chances of fiction or mistake 

 may be proportionately small, and in such cases we are satisfied 

 with ordinary testimony. 



Mr. J. S. Mill has some useful observations on Laplace's for- 

 mula in the chapter on the " Grounds of Disbelief" in his 

 1 System of Logic/ He draws attention to the absolute identity 

 supposed by hypothesis to exist between the 99 black balls, 

 which renders the case unlike that of real events. I have not 

 referred to this, because in fact it results from the nature of the 

 problem to be solved, in which we compare events of different 

 degrees of antecedent probability; the 99 black balls are not in- 

 tended to represent 99 similar events, but one and the same 

 event. For the purposes of calculation, the chances in favour 

 of an event must be treated as representing so many cases of its 

 occurrence. When we say that the chances are 9 to 1 in favour 

 of a certain horse A beating another horse B, there are only two 

 events conceivable, and only two sets of motives, &c. possible : 

 we do not conceive A as made up of 9 parts, each having an 

 equal chance of victory with B. We may speak, indeed, of 10 

 trials in which A will win 9 times ; but in each trial both horses 

 and both sets of motives are equally present. To express degrees 

 of probability, then, the most convenient method is to suppose 

 a proportionate number of identical events, as Laplace has done. 



