468 Mr. H. ^Vilbraham oh the Theory of Chances 



concur. Consequently those of the above sixteen compound 

 events which represent S occurring while any one or more of the 

 other three events do not occur, and which represent A, B, C all 

 to occur without S occurring, must be considered as beyond the 

 range of our observation. This does not contradict the former 

 assumption of the mutual independence of the four simple events; 

 for we do not by this last supposition say that such compound 

 events are impossible, nor do we make any new assumption as 

 to the probability of their occurrence, but only that, as they are 

 beyond the limits of our observation, we have nothing to do with 

 them. The events, therefore, which come within our circle of 

 observation are those marked respectively S, v, p, t, v, <^, ;^;, eo ; 

 and the absolute chance that any event which may occur is an 

 event within the range of our observation is 



xyzs-\-{l—x)yz[\—s)+x{\—]/)z{\.—s) + a;y{\ — s){\—s) 



+ x{l-y){\-z){l-s) + {\-x)y[\-z){\-s) 



+ [\-x){\-y)z{l-s) + [\-a^{\-y)[l-z)[\-s), 



which is similar to the quantity called V in Professor Boole's 

 book. 



I must here observe that x, y, and z are not the same as the 

 given quantities a, b and c; for the latter represent the chances 

 of A, B, and C respectively occurring, provided that the event 

 is one which comes within our range of observation, whereas 

 a.', y, and ;: represent the absolute chances of the same events 

 whether the event be or be not within that range. 



Of the eight events 8, v, p, r, v, (j), 'x^, o) which compose V, 

 four, viz. 8, p, T, and v, imply the occurrence of A. Consequently 

 the chance that if the event be within our range of observation 

 A will occur, is the sum of the chances of these last four events 

 divided by the sum of the chances of the eight. This will be 

 equal to the given chance a. Hence 



xyzs+ {{l-y)z + y{l-z) + {l-y){l-z)}x{l-s) _ 



V -"• 



So also 



xyzs + {{l-x)2 + x{l-z ) + {l-x){l-z)}t/{l-s) _ 



V ' " ' ~ 

 xyzs+{{\-y)z-r y{\-z) + (1 -y){l-z) ] z{\-s) _ 



V" 

 Also as the event S always in cases within our range of observa- 

 tion occurs conjointly with A, B and C, the chance of S occur- 

 ring and that of A, B, and C all occurring are the same, and 

 equal to m. Therefore 



xyzs 

 ~- =m. 



