528 Prof. G. Boole on the Theory of Probabilities, 



cation by common sense, to those who may choose to undertake 

 the task, will be the more easy. 



I Mill now exhibit the results to which this method conducts 

 us when applied to Mr. Mitchell's problem. That problem, 

 under a somewhat more general aspect, may be thus stated. 



Given the probability {p) of the truth of the proposition, If 

 the condition A is satisfied, the event B will not happen. 



Required the probability P of the ]iroposition. If the event 

 B does happen, the condition A has not been satisfied. The 

 result which I obtain is ' i^iO' 



p_ c{l-a) 



c{\-a)+a[\-pY 



where c and a are arbitrary constants, whose interpretation is as 

 follows: A'iz. a is the probability of the fulfilment of the con- 

 dition A, c is the probability that the event B would happen if 

 the condition X were not satisfied. 



Let us apply this solution to Mr. Mitchell's problem, and test 

 its agreement with common sense. The condition A is, that 

 the stars have been so distributed, that it is as likely that any 

 star will be found in one spot of the sky as another. Let us 

 term this a " random distribution," meaning thereby a distribu- 

 tion according to some law or manner, of the consequences of 

 which we should be totally ignorant ; so that it would appear to 

 us as likely that a star should occupy one spot of the sky as 

 another. Let us term any other principle of distribution an 

 indicative one. 



The event B is the occurrence somewhere in the heavens of a 

 double star as close as yS Capricorni. Hence p is the given proba- 

 bility that, on the principle of random distribution, there will not 



159 

 exist such a double star. Its numerical value appears to be -t^t^, 



80 1^^ 



°"'8l- 



P is the required probability, drawn from the existence of 

 /3 Capricorni, that the principle of random distribution did not 

 prevail ; a is the unknown probability of a random distribution, 

 c is the unknown probability, that, if the principle of random 

 distribution had not prevailed, such a star as (3 Capricorni would 

 have existed. 



If c and a could be determined in addition to jj, the value of 

 P would be definitely given by the formula*. 



* A cauilid mind will not object to this solution, the impossibility of 

 determining the unknown constants a and c by any actual exj)erience ; but 

 I can imagine such a mind as hesitating under the difficulty of conceiving 

 what kind of experience, could it be had, ^^•ould suffice for their detemiina- 

 tion. Perhaps the projicr answer would be, that sucli experience ought to 



