356 



Mr. A. Caylcy on a Question in the 



is assumed to be proportional to the given probability a of the 

 event A ; viz. the ratio in question is equal to that of the corre- 

 sponding sum for all the possible events to unity. 

 The entire series of possible events is 



A . AE . B . BE 

 A. AE.B'.(BE)' 

 A'.(AE)'.B.Be' 



(AE)' . (BE)' 



Boolian Probs. 



xsyt 



x s y't' 



x's'y t 



s't', 



which may be analysed as follows : viz. if AE and BE, then of 

 necessity A and B ; if AE and (BE)', then of necessity A and B' ; 

 if (AE)' and BE, then of necessity A' and B; but if (AE)' and 

 (BE)', then at pleasure A and B or A and B', or A' and B or A' 

 and B' : the sum proportional to unity therefore is 



xsyt + xsy't' 4- x's'yt + s't'. 



Now in the same manner as with a, dealing with the remaining 

 given quantities up, f3, @q, and with the required quantity u, we 

 have 



xsyt + xsy't' + xs't' 



xsyt + x's'yt + s'yt' 



_ xsyt + xsy't' 



up 

 __ xsyt 4- x's'yt 



__ xsyt 4- xsy't' 4- x's'yt 4- s't* 



_ xsyt -f xsy't' 4- x's'yt 

 ~ u ; 



(L) 



each of which is also 



x's'y'i' 



UB'W 



xsy't' 

 Al^E 



xs'y'i' 



aFI' 



x's'yt 



