182 Inverse Probability. [CHAP. vn. 



series of the desired kind. But we cannot say the same in 

 the case of the second example. 



12. The line of proof by which it is generally at 

 tempted to solve the second example is of this kind ; It is 

 shown that there being one white ball for certain in the bag, 

 the only possible antecedents are of ten kinds, viz. bags, 

 each of which contains ten balls, but in which the white 

 balls range respectively from one to ten in number. This of 

 course imposes limits upon the kind of terms to be found 

 in our series. But we want more than such limitations, we 

 must know the proportions in which these terms are ulti 

 mately found to arrange themselves in the series. Now this 

 requires an experience about bags which may not, and in 

 deed in a large proportion of similar cases, cannot, be given 

 to us. If therefore we are to solve the question at all we 

 must make an assumption ; let us make the following ; that 

 each of the bags described above occurs equally often, and see 

 what follows. The bags being drawn from equally often, it 

 does not follow that they will each yield equal numbers of 

 white balls. On the contrary they will, as in the last 

 example, yield them in direct proportion to the number of 

 such balls which they contain. The bag with one white 

 and nine black will yield a white ball once in ten times ; that 

 with two white, twice ; and so on. The result of this, it will 

 be easily seen, is that in 100 drawings there will be obtained 

 on the average 55 white balls and 45 black. Now with 

 those drawings that do not yield white balls we have, by the 

 question, nothing to do, for that question postulated the 

 drawing of a white ball as an accomplished fact. The series 

 we want is therefore composed of those which do yield white. 

 Now what is the additional attribute which is found in some 

 members, and in some members only, of this series, and 

 which we mentally anticipate ? Clearly it is the attribute of 



