580 Prof. G. Boole on the Theory of Probabilities. 



£(1:1^') >.^ 



c(l -«)+«(!-;,) P 



c{l — a)>c{\—a)p-\-ap{\-p), or c{\-a){\-p)>ap{\-p), 

 or c{\—a)>ap. Now 1 —a is the probability that the principle 

 of random distribution did not prevail^ and c is the probability 

 on this hj-pothesis that such a double star as /S Capricorni should 

 exist. Hence c(l —a) is the probability of the whole hypothesis, 

 that the distribution of the stars was not a random one, and that 

 double stars should exist. In like manner, ap is the probability 

 that the principle of random distribution did prevail, and that a 

 double star like /3 Capricorni should not exist. 



Hence we have the following conclusion : — If the probability 

 of an indicative law of distribution, and the consequent existence 

 of a double star, is greater than the probability in favour of a 

 random distribution, and a consequent absence of double stars, 

 then the probability in favour of an indicative law of tUstribution, 

 granting the existence of a double star, is stronger than the pro- 

 bability against a double star, granting the hyjjothesis of a dis- 

 tribution at random. 



If c and a were each equal to -, i. e. if it were a priori just as 



likely as not that the stars are scattered at random, and also just 

 as likely as not that, if they were not scattered at random, /9 Ca- 

 pricorni would exist, we should find 



1 



i 80 



P = 



1 1 ~ 81 



4 "^ 2 ^ 160 



I had intended to append to this paper an investigation of the 

 probability of the occurrence of triple stars of determinate close- 

 ness in the heavens, assuming Mr. MitchelFs law of distribution. 

 But the paper has already extended beyond its due limits, nor 

 would the investigation proposed at all serve the purpose of the 

 main argument. 



I trust that the design of this communication will not be mis- 

 understood. It is commonly the business of scientific methods 

 to teach us that which we could not learn without them ; but 

 to give us confidence in their results, it is also required that they 

 should, in those cases, upon which unassisted reason pronounces 

 a decision, not contravene her verdict. In the present instance 

 that confirmation has not been easy ; but it will, I trust, be 

 considered to have been complete. 



I am. Gentlemen, 

 Queen's CoUege, Cork, Your obedient Servant, 



March 11, 1851. Geoege Boole. 



