292 THE PRINCIPLES OF SCIENCE. 



probable hypothesis, I expect a recurrence of white balls. 

 The mathematician merely reduces this process of thought 

 to exact numbers. Taking, for instance, the hypothesis 

 that there are 99 white and one black ball in the urn, 

 he can calculate the probability that 20 white balls 

 should be drawn in succession in those circumstances ; he 

 thus forms a definite estimate of the probability of this 

 hypothesis, and knowing at the same time the probability 

 of a white ball reappearing if such be the contents of the 

 urn, he combines these probabilities, and obtains an exact 

 estimate that a white ball will recur in consequence of 

 this hypothesis. But as this hypothesis is only one out 

 of many possible ones, since the ratio of white and black 

 balls may be 98 to 2, or 97 to 3, or 96 to 4, and so on, 

 he has to repeat the estimate for every such possible 

 hypothesis. To make the method of solving the problem 

 perfectly evident, I will describe in the next section a 

 very simple case of the problem, originally devised for the 

 purpose by Condorcet, which was also adopted by Lacroix !l , 

 and has passed into the works of De Morgan, Lubbock, 

 and others. 



Simple Illustration of the Inverse Problem. 



Suppose it to be known that a ballot-box contains only 

 four black or white balls, the ratio of black and white balls 

 being unknown. Four drawings having been made with 

 replacement, and a white ball having appeared on each 

 occasion but one, it is required to determine the proba- 

 bility that a white ball will appear next time. Now the 

 hypotheses which can be made as to the contents of the 

 urn are very limited in number, and are at most the 

 following five : 



R 'Trait dldmentairc du Calcul des Probability' 3rd ed. (1833), 

 p. 148. 



