developed in Professor Boole's " Laws of Thought." 471 



his reasoning " does not require that the drawings of a white and 

 marble ball should be independent in virtue of the physical con- 

 stitution of the balls ; that the assumption of their independence 

 is indeed involved in the solution, but does not rest upon any 

 prior assumption as to the nature of the balls, and their relations 

 or freedom from relations, of form, colour, structure, &c." 

 (page 262), I am at a loss to understand. It would appear 

 that its being involved in the solution proves that it must rest 

 on a prior assumption, and that the prior assumption in this case 

 is that the simple events are subject to the results of Prop. I. 



When additional conditions, that the chance of a combination 

 of events (\>^[x, y, . . . ) is m, that of ^ci{x, y,. . .) is n, and so on, 

 are given, and consequently subsidiary events s, t are introduced, 

 the question becomes this : — given that x, y, . . . s,t, . . . are in- 

 dependent events, and that if it be known that the event which 

 will happen will belong to a certain group of events selected out of 

 the whole number of possible combinations, in which s does not 

 happen except in conjunction with <^^{x,y, . . .), nor ^^{x, y,...) 

 except in conjunction with s, and so on with respect to t and 

 ^^{x, y,...), &c., the chances of x, y,. . . are p, q, . . . and of 

 s,t, ... are m,n, ...; required the absolute probabilities oi x,y,... 

 when we have no such pre\Tious knowledge; or more usually, 

 required the probability that out of the same group of events as 

 before the event will be some definite combination oi x,y,. . . 



The independence of the events x,y, .. .s, t, . . . is, as before, 

 assumed in the assumption of the results of Prop. I. Never- 

 theless Professor Boole says (page 264) that the events denoted 

 by s, t, &c., whose probabilities are given, have such probabilities 

 not as independent events, but as events subject to a certain con- 

 dition V. He seems throughout to consider V as a condition 

 which does always obtain, and consequently that the chance of 

 any event inconsistent with it is 0, and therefore he ignores the 

 previously assumed independence of the simple events which is 

 inconsistent with such a supposition, instead of considering V as 

 a condition which, if it obtain, the chances of x, y, . . . are as 

 given in the data of the problem. 



I will now take the first problem of Chap. XX. p. 321, which 

 is the question treated of by Mr. Cayley in a paper in the Philo- 

 sophical Magazine of last October, which elicited an answer from 

 Professor Boole in a succeeding Number of the same Magazine, 

 and work it out in the same manner as I have done a former 

 question on Professor Boole's assumptions. The question is, — 

 the probabilities of two causes A, and Aj are c, and Co respect- 

 ively ; tlic probability that if Aj happen E will happen is p^, that 

 if Aj happen E will happen is p^. E cannot happen if neither 

 A, nor A^ happen. Required the probability of E. 



