CHAP. XX.] PROBLEMS ON CAUSES. 333 



form, whether we take account in them of the occurrence or of 

 the non-occurrence of E, it is evident that the solution ought to 

 be, as it is, a symmetrical function of p, q and 1 - p 9 I - q, 



Let us examine the particular case in which p = 1, q = 1. 

 We find 



h = a + b - 1, h f = - 1, / = ab, m = a -f b - 2, 

 and substituting 



Prob. x - 2a (a + b - 1) 



-2ab(a + b-l) 

 -2a(a + b-l)~ 



It would appear, then, that in this case the events A and B are 

 virtually independent of each other. The supposition of their 

 invariable association with some other event E, of the frequency 

 of whose occurrence, except as it may be inferred from this par- 

 ticular connexion, absolutely nothing is known, does not establish 

 any dependence between the events A and B themselves. I ap- 

 prehend that this conclusion is agreeable to reason, though par- 

 ticular examples may appear at first sight to indicate a different 

 result. For instance, if the probabilities of the casting up, 1st, 

 of a particular species of weed, 2ndly, of a certain description of 

 zoophytes upon the sea-shore, had been separately determined, 

 and if it had also been ascertained that neither of these events 

 could happen except during the agitation of the waves caused by 

 a tempest, it would, I think, justly be concluded that the events 

 in question were not independent. The picking up of a piece of 

 seaweed of the kind supposed would, it is presumed, render more 

 probable the discovery of the zoophytes than it would otherwise 

 have been. But I apprehend that this fact is due to our know- 

 ledge of another circumstance not implied in the actual conditions 

 of the problem, viz., that the occurrence of a tempest is but an 

 occasional phenomenon. Let the range of observation be con- 

 fined to a sea always vexed with storm. It would then, I sup- 

 pose, be seen that the casting up of the weeds and of the 

 zoophytes ought to be regarded as independent events. Now, 

 to speak more generally, there are conditions common to all phae- 



