xii.] THE INDUCTIVE OK INVERSE METHOD. 255 



we may add together these separate probabilities, and we 

 find that 



81 , 32_ , 3 116 



184 + 184 + 184 or T8^ 



is the complete probability that a white ball will be next 

 drawn under the conditions and data supposed. 



General Solution of the Inverse Problem. 



In the instance of the inverse method described in the 

 last section, the balls supposed to be in the ballot-box 

 were few, for the purpose of simplifying the calculation. 

 In order that our solution may apply to natural phe- 

 nomena, we must render our hypotheses as little arbitrary 

 as possible. Having no a priori knowledge of the con- 

 ditions of the phenomena in question, there is no limit 

 to the variety of hypotheses which might be suggested. 

 Mathematicians have therefore had recourse to the most 

 extensive suppositions which can be made, namely, that 

 the ballot-box contains an infinite number of balls ; they 

 have then varied the proportion of white to black balls 

 continuously, from the smallest to the greatest possible 

 proportion, and estimated the aggregate probability which 

 results from this comprehensive supposition. 



To explain their procedure., let us imagine that, instead 

 of an infinite number, the ballot-box contains a large 

 finite number of balls, say 1000. Then the number of 

 white balls might be I or 2 or 3 or 4, and so on, up to 

 999. Supposing that three white and one black ball 

 have been drawn from the urn as before, there is a certain 

 very small probability that this would have occurred in 

 the case of a box containing one white and 999 black 

 balls ; there is also a small probability that from such a 

 box the next ball would be white. Compound these 

 probabilities, and we have the probability that the next 

 ball really will be white, in consequence of the existence 

 of that proportion of balls. If there be two white and 998 

 black balls in the box, the probability is greater and will 

 increase until the balls are supposed to be in the propor- 

 tion of those drawn. Now 999 different hypotheses are 

 possible, and the calculation is to be made for each of 

 these, and their aggregate taken as the final result. It is 



