644 PROFESSOR BOOLE ON THE COMBINATION 



the values which they respectively give to the probability of the event z are / \ 

 Pz . . p„. The result is 



(Pi Pi • • Pn ) » 



r= . ^7 '- ri • (1) 



\PlPi • • Pn)" + (i 1 -Pi) 0--P-2) ■ • (t-Pn) ) " 



This is the general formula of the mean in reference to judgments, and much as 

 it differs from the formula of the mean, in reference to the observations of a 

 physical magnitude, some remarkable points of analogy exist. I will notice but 

 one. The arithmetical mean is not altered if to the quantities among which it is 

 taken we add another equal to the previous mean. Thus we have 



Pl+Pl-- +Pn + 1 _ Pl+Pt • ■ + Pn 



n + 1 n 



provided that^»+i = Pl +P2 ' ' +Pn ' Or representing ^ l + p -' + ^ by P n we have 



P — P 

 provided that 



P*+i=?« .... (2) 



The same relation may readily be shown to hold also, if P„ represent the mean of 

 judgment, as expressed in (1). 



39. The following is a brief summary of the conclusions established in this 

 paper. 



ls£, The solution of the problem of astronomical observations by the logical 

 theory of probabilities is, in its general form, indefinite. 



2ndly, It becomes definite, if we introduce the general principle of means. The 

 result is in accordance with the usual formulae, but expresses the so-called weights 

 of the observations as determinate functions of certain probabilities relating to 

 the correctness of the observations, and the character of the observers. 



3dlz/, When, as respects the two last elements, the observations are considered 

 equal, the formula is reduced to the expression of the arithmetical mean. 



Stilly, The complete solution of the problem of the combination of two proba- 

 bilities of an event founded upon different testimonies or judgments is indefinite, 

 but admits, in various cases, of being reduced to a definite form. 



bthly, This indefiniteness is due to the circumstance indicated by the formula, 

 that the strength of the probabilities in combination is due, not to the strength 

 of the separate probabilities alone, but also to the degree of unexpectedness of 

 the testimonies or judgments themselves. 



Qtlily, Combined presumptions, whether for or against an event, are generally 

 strengthened by the unexpectedness of the combination. 



"tidy, When probabilities as p v p 2 , . .p n are in a high degree cumulative, owing 



