366 PROBLEMS ON CAUSES. [CHAP. XX. 



Let xy = t, then 



-^ (3) 



Hence observing that Prob. x = a, Prob. t = ap, and passing from 

 Logic to Algebra, we have 



tx + cl -t x 



Frob ^ 



with the relations 



op 



Hence we readily find 



Prob. y = op + c (1 - a). (4) 



Now recurring to (3), we find that c is the probability, that if 

 the event (1 - f) (1 - x) occur, the event y will occur. But 



Hence c is the probability that if the event x do not occur, 

 the event y will occur. 



Substituting the value of Prob. y in (2), we have the follow- 

 ing theorem : 



The calculated probability of any phenomenon y, upon an as- 

 sumed physical hypothesis x, being p, the a posteriori probability P 

 ofthephysical hypothesis, when the phenomenon has been observed, 

 is expressed by the equation 



where a and c are arbitrary constants, the former representing the 

 a priori probability of the hypothesis, the latter the probability that 

 if the hypothesis were false, the event y would present itself. 



The principal conclusion deducible from the above theorem 

 is that, other things being the same, the value of P in creases and 

 diminishes simultaneously with that of p. Hence the greater or 

 less the probability of the phenomenon when the hypothesis is 

 assumed, the greater or less is the probability of the hypothesis 

 when the phenomenon has been observed. When p is very small, 

 then generally P also is small, unless either a is large or c small. 



