372 BISHOP TERROT ON PROBABILITIES. 



of the effects reasonably produced upon the minds of the informants, by the know- 

 ledge of certain facts which they have not communicated to us. The fractions 

 which they give are admitted as true exponents of the results of their respective 

 partial knowledge, no doubt resting either upon their veracity, or upon the accu- 

 racy of their inferences. We admit, that each states the probability as he ought, 

 under his circumstances : and the question is, how ought we to state it under our 

 circumstances, knowing as we do something more, and also something less than 

 either of our informants. 



(6.) In attempting to answer this question, I shall have recourse to the ordi- 

 nary illustration of an urn and balls. Let us suppose that A has seen p white 

 and q-p black balls introduced into an urn, which he believes to have been pre- 

 viously empty. He properly infers that the probability of drawing a white is 



J -. B, under the same circumstances, has seen r white, and s-r black balls in- 



'1 



troduced, and infers that the probability of drawing a white is -. If they com- 

 municate to each other only their inferences, there is an apparent contradiction, 

 and no combination or agreement can take place. But if they communicate the 

 facts from which the inferences were deduced, then each knows that the urn con- 

 tains p + r white, and q + s-p—r black balls, and agree in making the probability of 



drawing a white ^— - . If the number of balls whose introduction has been seen 

 by the two observers be equal, then P±L-P±± = (P + l\ = the sum of the 



J ^ q + s 2 q 2 \q q) 2 



several probabilities. 



try _L *» 



It may be observed that —_ — , as the expression of the combined probabilities 



- and -, is not exposed to the objection of admitting contradictory results, for, 

 if we take the negative as the conclusion whose probability is to be found, then 

 A gives for the probability of the conclusion il£=2~P while B gives 1--=— r , 



therefore the combined probability against the event is q ~ p s ~ r . But the 

 combined probability for the event was ^- and ^±1 + q ~P + s ~ r = l±l = i ? as j t 



ought to be. 



(7.) But what we have to consider, is the impression made upon the mind of 

 a third person, who is informed by A that, from his observation, the probability 



of drawing a white is -, and by B that, from his observation, it is -, and to 



whom no farther information is given, except that the observations were totally 

 distinct. Now, as these data give only the ratio of white to black balls at each 

 introduction, there may have been, in the first, p white and q—p black, or there 



