470 Mr. H. Wilbraham on the Theory of Chances 



the independence of only three events, which is eqiuvalent to 

 assuming four equations ; and when an additional datum was 

 given^ viz. the chance m of the concurrence of A, B and C, we 

 assumed the independence of four events A, B, C and S, which 

 implies eleven equations^ whereas we might have expected that 

 one assumed equation less than before would have been requi- 

 site ? The answer is, that though all the eleven equations have 

 been stated to be assumed, only some of them are in the actual 

 working of the problem necessary assumptions. It is sufficient 

 that the eleven equations should be true so far as they affect the 

 relations among the eight contingencies in the compound event 

 represented by V. It will be found that three only out of the 

 eleven give such relations ; and upon the assumptions comprised 

 .in these last three equations rests the truth of the solution. The 



three equations are —=- = —, and ^=— . The other eight 



o> X « % 



equations, though not contradictory to the data, are not essential 

 to the solution, and need not have been assumed. If these three 

 conditions had been inserted in the data of the problem, it might 

 have been solved by a simple algebraical process without intro- 

 ducing the subsidiary event S. 



This assumption of the independence of the simple events 

 made directly in the solution I have given of the last question, 

 is, as I have said, tacitly made in the logical solutions of the 

 questions given in Professor Boole's book. In Projiosition I. 

 of Chapter XVII. the events represented by x, y, &c. are by 

 hypothesis independent. In other words, the equations of 

 condition implied by that independence (in number, 1 if there 

 be 2 events ^, y, 4 if there be 3 events, 2"— n— 1 if there be n 

 events) are assumed to subsist among the compound events, 

 which are combinations of the simple events x, y, &c. The 

 theorem is proved and proveable only on this assumption. This 

 proposition is assumed in Prop. II., and forms the basis of the 

 application of the logical equations to questions of chances. In 

 Prop. II. p. 261, the question is of this nature; given that, 

 whenever it be known that the event which will happen will 

 belong to a certain group of events represented by V, the chance 

 of X happening is p, of y, q, &c. ; required the absolute probabi- 

 lities oi X, y , . . . when we have no such previous knowledge. 

 As in the solution in the book, Prop. I. is in the outset assumed 

 with regard to x, y, . . . , the conditions of Prop. I. are assumed, 

 and one of these is that x, y, . . . are " simple unconditioned 

 events,'' which (page 258) implies that they arc independent. 

 Consequently x, y, . . . are in Prop. II. assumed to be indepen- 

 dent. How this can be I'cconciled with Professor Boole's state- 

 ment with regard to ^ particular example of the proposition that 



