CHAP. XX.] PROBLEMS ON CAUSES. 369 



probabilities which rest upon insufficient data, is to assign to some 

 element whose real probability is unknown all possible degrees 

 of probability ; to suppose that these degrees of probability are 

 themselves equally probable ; and, regarding them as so many dis- 

 tinct causes of the phenomenon observed, to apply the theorems 

 which represent the case of an effect due to some one of a number 

 of equally probable but mutually exclusive causes (Problem 9). 

 For instance, the rising of the sun after a certain interval of 

 darkness having been observed m times in succession, the proba- 

 bility of its again rising under the same circumstances is deter- 

 mined, on received principles, in the following manner. Let p 

 be any unknown probability between and 1 , and c (infinitesimal 

 and constant) the probability, that the probability of the sun's 

 rising after an interval of darkness lies between the limits p and 

 p + dp. Then the probability that the sun will rise m times in 

 succession is 



\p m dp-, 



J 



and the probability that he will do this, and will rise again, or, 

 which is the same thing, that he will rise m + 1 times in succes- 

 sion, is 



c 



'0 



Hence the probability that if he rise m times in succession, he will 

 rise the m + 1 th time, is 



c{ p m + l dp 

 V F m + * 



f 1 ~ m + 2 ' 



c I p m dp 



the known and generally received solution. 



The above solution is usually founded upon a supposed analogy 

 of the problem with that of the drawing of balls from an urn con- 

 taining a mixture of black and white balls, between which all 

 possible numerical ratios are assumed to be equally probable. 

 And it is remarkable, that there are two or three distinct hypo- 

 theses which lead to the same final result. For instance, if the 

 balls are finite in number, and those which are drawn are not 



2B 



