472 Mr. H. Wilbraham on the Theory of Chances 



I will first, however, examine what result can be arrived at 

 without making: any assumption. Let ^ be the chance of A, and 

 A.2 both happening and being followed by E, ^'that of their both 

 happening but not followed by E, rj and t}' the chances of Ag 

 happening without A,, according as it is followed and not fol- 

 lowed by E, ^aud ^' those of A, happening without Ag accord- 

 ing as it is followed or not by E, and cr' the chance of neither Aj 

 nor Ao happenings and E of course not happening. The data of 

 the problem give the equations 



^+^' + V+V' = C2 



the chance (m) of E happenin g = |^ + ^; + ?= c, ja^ + CgjSg — f , where f 

 is necessarily less than either c^p^ or c^pq. We can get no further 

 in the solution without further assumptions or data, having only 

 six equations from which to eliminate seven unknown quantities. 

 Without such the question is indeterminate. 



Now, to adopt Professor Boole^s assumptions, let x, y, z be 

 the chances of A,, Ag, and E respectively, and s, t those of two 

 subsidiary events ; x, y, s, t are assumed to be mutually inde- 

 pendent events; consequently the chances of the sixteen mu- 

 tually exclusive contingencies formed by combinations of these 

 four simple events will be 



x{\-y){\-s)t, 



x{\-y)s{\-t), 



xy{\-s)[\-t), 



x{\-y){\-s){l-l), 



{l-x)y{\-s){\-t), 



(l-a.)(l-yMl-0, 



{\-x){\-y){\-s)t, 



{l-:,){l-y){l-s){\-t). 



The relations among these sixteen events implied by the inde- 

 pendence of the four simple events are, as before, eleven in 

 number. As the events represented by s and t in all cases 

 within our range of observation are concomitant with the con- 

 currence of Aj and E, and of A^ and E respectively, the events 

 represented by e, 6, i, k, \, v,-)(^ts,'>^ must be struck out, being 

 inconsistent with such concomitance, and consequently the ag- 

 gregate event V comprises only the events 8, /x, p, t, u, (j>, a>. 



