developed in Professor Boole's " Laws of Thought." 467 



chances besides the absolute chances of the simple events A 

 and B. The new ffiven condition appears now to supersede and 

 take the place of the previously assumed condition ; and with this 

 new condition combined with the three former equations among 

 6, X, /x, (f), the problem is easily solvable by algebra. In the 

 second question, where there are three simple events A, B and C, 

 suppose there to be one other given relation among the chances. 

 This new condition certainly does to some extent supersede those 

 previously assumed ; and it appears to me that Professor Boole's 

 reasoning would lead one to suppose that the former assumptions 

 are entirely banished from the problem, and no others except the 

 said newly given condition assumed in their stead. The fact, 

 howevei', is that in this case certain additional assumptions are 

 made, otherwise the problem would be indeterminate. The 

 nature of these assumptions, which are different from the assump- 

 tions made when no condition besides the absolute chances of 

 the simple events is given, will perhaps be better seen from the 

 following discussion of an example than from any general rea- 

 soning. I shall adopt in it the same assumptions as are made 

 in Professor Boole's method, but work it out without the aid of 

 his logical equations. Any question which can be solved by the 

 logical method may also be treated in this manner. 



The chances of three events A, B, and C are a, b, c respect- 

 ively, and the chance of all three happening together is m ; what 

 is the chance of A occurring without B ? 



Suppose A, B and C, and a fiu-ther event S, to be four simple 

 events mutually independent, the absolute chances of which are 

 respectively x, y, z and s. We suppose for the present no con- 

 nexion to exist between the original simple events A, B and C, 

 and the subsidiary event S. There will be altogether sixteen 

 possible mutually exclusive compound events, the chances of 

 which (since the simple events are independent) are as follows :• — 



Let us now make an assumption with respect to the subsidiary 

 event S, viz. that it is never observed except in conjunction with 

 the three other events, and is always observed to happen if they 



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