APPROXIMATE GENERALIZATIONS. 133 



every three As are Bs, and three of every four Cs are Bs, the 

 probability that something which is both an A and a C is a B, 

 will be more than two in three, or than three in four. Of 

 every twelve things which are As, all except four are Bs by the 

 supposition ; and if the whole twelve, and consequently those 

 four, have the characters of C likewise, three of these will be Bs 

 on that ground. Therefore, out of twelve which are both As 

 and Cs, eleven are Bs. To state the argument in another way ; 

 a thing which is both an A and a C, but which is not a B, is found 

 in only one of three sections of the class A, and in only one of 

 four sections of the class C ; but this fourth of C being spread 

 over the whole of A indiscriminately, only one-third part of it 

 (or one-twelfth of the whole number) belongs to the third section 

 of A ; therefore a thing which is not a B occurs only once, 

 among twelve things which are both As and Cs. The argu- 

 ment would in the language of the doctrine of chances, be thus 

 expressed : the chance that an A is not a B is ^, the chance 

 that a C is not a B is ^ ; hence if the thing be both an A and 

 a C, the chance is ^ of ^ = T V 



It has, however, been pointed out to me by a mathematical 

 friend, that in this statement the evaluation of the chances is 

 erroneous. The correct mode of setting out the possibilities 

 is as follows. If the thing (let us call it T) which is both an 

 A and a C, is a B, something is true which is only true twice 

 in every thrice, and something else which is only true thrice 

 in every four times. The first fact being true eight times in 

 twelve, and the second being true six times in every eight, and 

 consequently six times in those eight ; both facts will be true 

 only six times in twelve. On the other hand if T, although it is 

 both an A and a C, is not a B, something is true which is 

 only true once in every thrice, and something else which is 

 only true once in every four times. The former being true 

 four times out of twelve, and the latter once in every four, and 

 therefore once in those four ; both are only true in one case 

 out of twelve. So that T is a B six times in twelve, and T is 

 not a B, only once : making the comparative probabilities, not 

 eleven to one, as I had previously made them, but six to one. 



It may be asked, what happens in the remaining cases? 



