THE INDUCTIVE OR INVERSE METHOD. 293 



4 white and o black balls 



> 3 



4 



The actual occurrence of black and white balls in the 

 drawings renders the first and last hypotheses 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 f x f x f x i, by 

 the rule already given (p. 233). But as it is indifferent 

 to us in what order the balls are drawn, and the black 

 ball might come first, second, third, or fourth, we must 

 multiply 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 1 x l x l x 4, or g-J, and from the 

 third hypothesis of one white and three black we deduce 

 likewise ^ x x ^ x f x 4, or f^. According, then, as we 

 adopt the first, second, or third hypothesis, the proba- 

 bility that the result actually noticed would follow is ff, 

 ~, and 6 \. Now it is certain that one or other of these 

 hypotheses must be the true one, and their absolute 

 probabilities are proportional to the probabilities that the 

 observed events would follow from them (see p. 279). All 

 we have to do, then, in order to obtain the absolute pro- 

 bability of each hypothesis, is to alter these fractions in 

 a uniform ratio, so that their sum shall be unity, the 

 expression of certainty. Now since 27 + 16 + 3 = 46, 

 this will be effected by dividing each fraction by 46 and 



