350 PROF. W. STANLEY JEVONS ON 



when B does as when B docs not ; in short,, that they are 

 independent events. As a case where we are not to assume 

 logical independence^ we may take the following example 

 from Boole^s work (p. 276) : — 



Example i. '^^The probability that it thunders upon a 

 given day is p, the probability that it both thunders and 

 hails is q-, but of the connexion of the two phenomena of 

 thunder and hail nothing further is supposed to be known. 

 Required the probability that it hails on the proposed 

 day." 



Let A mean that it thunders 



B „ ,, hails; 



Then there are four possible events, AB, A6, oB, ah. 

 The probabilities given are — 



Prob. of A =p, 

 „ „ AB^^'. 



The probability required is that of B, which is evidently 



prob. of AB -f prob. of aB. 



Now the probability of AB is given, but the probability of 

 aB is not given, and we cannot assume it to be (i— jf?) 

 X (prob. of B), because we are told that nothing is known 

 of the connexion of the phenomena, which implies that 

 they may have some connexion by causation, so that the 

 non-occurrence of A will alter the probability of the oc- 

 currence of B. The prob. of «B is therefore unknown, 

 except that it is the prob. of a multiplied by the unknown 

 prob. that, if a occurs, B occurs with it, as Boole points 

 out. Hence the only possible answer is the same as 

 Boole^s 



prob. of B = 5' + ( I —p) Cy 



c being an unknown quantity, of course not exceeding 

 unity. Making c successively = i and o, the major and 



