370 PROBLEMS ON CAUSES. [CHAP. XX. 



replaced, or if they are infinite in number, whether those drawn 

 are replaced or not, then, supposing that m successive drawings 

 have yielded only white balls, the probability of the issue of a 

 white ball at the m + I th drawing is 



m 



m + 2 ' 



It has been said, that the principle involved in the above 

 and in similar applications is that of the equal distribution of 

 our knowledge, or rather of our ignorance the assigning to 

 different states of things of which we know nothing, and upon 

 the very ground that we know nothing, equal degrees of proba- 

 bility. I apprehend, however, that this is an arbitrary method of 

 procedure. Instances may occur, and one such has been adduced, 

 in which different hypotheses lead to the same final conclusion. 

 But those instances are exceptional. With reference to the par- 

 ticular problem in question, it is shown in the memoir cited, that 

 there is one hypothesis, viz v when the balls are finite in number 

 and not replaced, which leads to a different conclusion, and it is 

 easy to see that there are other hypotheses, as strictly involving 

 the principle of the "equal distribution of knowledge or igno- 

 rance," which would also conduct to conflicting results. 



24. For instance, let the case of sunrise be represented by 

 the drawing of a white ball from a bag containing an infinite 

 number of balls, which are all either black or white, and let the 

 assumed principle be, that all possible constitutions of the system 

 of balls are equally probable. By a constitution of the system, I 

 mean an arrangement which assigns to every ball in the system 

 a determinate colour, either black or white. Let us thence seek 

 the probability, that if m white balls are drawn in m drawings, 

 a white ball will be drawn in the m + I th drawing. 



First, suppose the number of the balls to be /*, and let the 

 symbols 19 # 2 , . . M be appropriated to them in the following 

 manner. Let X L denote that event which consists in the i th ball 

 of the system being white, the proposition declaratory of such a 

 state of things being xi = 1. In like manner the compound 



* See a memoir by Bishop Terrot, Edinburgh Phil. Trans, vol. xx. Part iv. 



