xii.J THE INDUCTIVE OR INVERSE METHOD. 253 



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, 1 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 : 



4 white and o black balls 



3 i 



2 ,,2 



i ,,3 

 o ,,4 



The actual occurrence of black and white balls in the 

 drawings puts the first and last hypothesis out of the 

 question, so that we have only three left to consider. 



If the box contains three white, and one black, the 

 probability of drawing a white each time is f , and a black 

 ^ ; so that the compound event observed, namely, three 

 white and one black, has the probability X f X f X ^, by 

 the rule already given (p. 204). But as it is indifferent 

 in what order the balls are drawn, and the black ball 

 might come first, second, third, or fourth, we must multi- 

 ply by four, to obtain the probability of three white and 

 one black in any order, thus getting f^-. 



Taking the next hypothesis of two white and two 

 black balls in the urn, we obtain for the same proba- 

 bility the quantity |x|x^xjx4, or , and from the 

 third hypothesis of one white and three black we deduce 

 likewise x x x f x 4, or U 3 f . According, then, as we 



1 Trait* iUmentaire du Calcul des Probalilites, 3rd ed. (1833), 

 p. 148. 



