on some Questions in the Theory of Chances. 153 



The method which was used for summing' the series in the 

 last page is of very general application, and depends in fact on 

 this principle, that the generating function of the sum of any 

 series is the sum of the generating functions of each of the terms 

 of the series. 



If in the last formula, «, > "3 , &c. = 0, and if there be only two 

 events possible, and w, = 1, the probability required is 



1 + 6'e, 

 m+ 2 ' 



In order to apply this, suppose an individual to have asserted 

 m events to have taken place, of which the simple probabilities 

 are equal and equal to p, and suppose it required to find the 

 probability of his telling the truth in another case, where the 

 simple probability of the event he asserts to have taken place 

 is not known. Let x be the veracity of the individual, the pro- 

 bability of his telling the truth on this hypothesis is 



P£ . 



px + (1 -x) (1 -p)' 



and the probability of his telling the truth is the sum of the 

 probabilities of his telling the truth on each hypothesis, divided 

 by the number of the hypotheses. 



Suppose X to vary from to 1, and all these values of x to 

 be equally probable a priori, then the probability of his having 

 told the truth and the event having taken place, is 



/; 



pxdx 



px +' (1— p) {y—x) ' 

 taken from x = to x = 1 , which integral is 



2J5-1 V 



9 



^_(i_iij:Z2hyp. log. -^1 



2J5-1 \ 2p—l ^^ ^ 1 -p\ 



I( p = j- , this probability is .81601, generally if p >f, the 

 assertion that the event has taken place (on this hypothesis of 



Vol. III. Fart I. U 



