132 On the Theonj of Association 



etc. miist first be tried ; if thcy are zero, then follow on with | AB \C\,\ AB \D\, 

 ! AC\ D 1 and \BG\ D\, but not . AC\ B \ or \ AD j B \ etc. ; if they arc zero then 

 finally try j AB \ CD \, if it also be zero tlien the attributes are completely inde- 

 pendent. It is not necessary to try further | AG\ BD \ or | AD \ BC \ etc. 



The inadequacy of the usual treatment of indepondence ariscs from the fact 

 that it proceeds wholly d priori, and gcnerally has reference solely to cases of 

 artificial chancc. The rcsult is an ilhisoiy appearance of siiiiplicity. It is pointed 

 out that if ono "evcnt" can "succeod" in «i and " fail " in ^, ways, a second 

 succeed in cu and fail in h^ ways, and so on, the combined evonts can take place 

 (succeed or fail) in 



(«i + ^)(a, + L)...(a„ + 6„) 

 ways and succeed in 



ttlf/o ... «„ 



ways. The chance of entire "success" is therefore 



OiCfcj ... On 



(ai + 6i)(a, + fe,)...(((„ + 6„)' 

 the chance of the first event failing and the rest succeeding is 



hiür, ... a„ 



and so on for all other possible cases. In short the chance of occurrence of the 

 combined indepcudent events is the product of the chances of the separate 

 events. There the treatment stops, and all practical difficulties are avoided. In 

 such text-book treatment it is given that the events are independont and required 

 to deduce the conscquences ; in the problems that the statistician has to handle 

 the consequences — the bare facts — are given and it is required to find whcther 

 the "events" or attributes are independent, wholly or in part. 



5. On the fallacies that maij he caused hy the mixiiig uf distinct records. 



It foUows from the preceding work that \ve cannot infer indcpendence of a 

 pair of attributes within a sub-universe from the fact of indepondence within the 

 universe at largo. From \ AB \ = 0, we cannot infer | AB j C | = or | AB | 7 | = 0, 

 although we can of course make the corresponding inference in the case of 

 complete association — i.e. from | AB | = 1 we do infer \ AB\G\ = \ AB 1 7 [ = etc. = 1. 

 But the convcrse theorem is also true ; a pair of attributes does not neces.sarily 

 exhibit indepondence within the universe at large even if it cxhibit indepondence 

 in every sub-universe ; given 



|.4B|C| = 0, 1 .45 1 7 1 = 0, 



we cannot infer |.45| = 0. Thr thooroni is of considerable practical importance 

 from its invcrse application ; i.e. cven if |.(1B| have a sensible positive or 



