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XXV. — On the Possibility of combining two or more Probabilities of the same Event, 

 so as to form one Definite Probability. By the Right Rev. Bishop Terrot. 



(Read 17th March 1856.) 



(1.) The inquiry which, with its results, I propose to lay before the Society, 

 was suggested by the following passage in the very popular Treatise on Logic by 

 Dr Whately, now Archbishop of Dublin. 



" As in the case of two probable premises, the conclusion is not established 

 except upon the supposition of their being both true, so in the case of two (and 

 the like holds good with any number) distinct and independent indications of the 

 truth of some proposition, unless both of them fail, the proposition must be true : 

 we therefore multiply together the fractions indicating the probability of the 

 failure of each — the chances against it — and the result being the total chances 

 against the establishment of the conclusion by these arguments, this fraction be- 

 ing deducted from unity, the remainder gives the probability for it. 



" E. g. A certain book is conjectured to be by such and such an author, partly, 

 1st, from its resemblance in style to his known works ; partly, 2d, from its being 

 attributed to him by some one likely to be pretty well informed. Let the proba- 

 bility of the conclusion, as deduced from one of these arguments by itself, be sup- 



2 3 



posed ^, and in the other case = ; then the opposite probabilities will be respec- 



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tively -= and =, which multiplied together give ^ as the probability against the 



conclusion ; i. <?., the chance that the work may not be his, notwithstanding the 

 reasons for believing that it is ; and, consequently, the probability in favour of 



23 2 " 



the conclusion will be — , or nearly^ (Whately's Logic, 8th Ed., p. 211.) 



(2.) Now, this reasoning appears to me erroneous, because it can be so applied 

 as to bring out two inconsistent conclusions. It must be observed, that there is 

 no such generic difference between the chances for and against the truth of a pro- 

 position, as can require or justify any difference in the laws and methods applied 

 to them. A negative can always be turned into an affirmative by a change of 

 verbal expression, without any change of meaning. Thus the chance of not hit- 

 ting a mark is the same as the chance of missing it. The chance of a life not fall- 

 ing before sixty, is the chance of its continuance up to sixty. The chance that A 

 was not the author of the book, is the chance that some one else was the author. 



Let us then take as the proposition whose probability is to be found, the negative 



— he did not write it — the partial probabilities for which are by the data - and =' 



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VOL. XXI. PART III. 5 H 



