DEVIATION OF R.-INPOM S.tMPI.ES 93 



1 — The bag contains 1 white and 999 not white. 

 2 — The bag contains 2 white and 998 not white. 

 3 — The bag contains 3 white and 097 not white. 



A' — The bag contains K white and {l,OO0-K) not white. 



997 — The bag contains 997 white and 3 not white. 

 998 — The bag contains 998 white and 2 not white. 

 999 — The bag contains 999 white and 1 not white. 



Let Tr(A') be the existence probability for the A"th hypothesis. 

 By "existence probabiHty" is meant the likelihood that the bag 

 contains exactly A white balls when the circumstances of the drawing, 

 but not the actual results of the drawing, are fully taken into account. 

 Its exact value may often be in doubt either because we do not have 

 complete knowledge of the circumstances preceding the drawing 

 or because we are not able to deduce its exact value from this knowl- 

 edge. It is obvious, however, that there must be some such value 

 and we must, therefore, introduce a symbol to represent it. 



Let 5(7, 100, A) = productive probability for the A'th hypothesis; 

 by this is meant the probability of obtaining the observed event (7 

 white in 100 drawings) if the bag contains A white balls and 1,000-A 

 that are not white. 



Then the a posteriori probability in favor of the A'th hypothesis 

 (meaning thereby the probability in favor of the A'th hypothesis 

 after the 7 white balls were drawn) is * 



W{K)B{7,100,K) 

 2 W(5)B(7,100, S) 



Now to say that the bag with a total of 1,000 balls contains A white 

 balls is equivalent to saying that the ralio of white to total balls is 



pk=K /lOOO 



and that the ratio of not white to total balls is 



54 = 1-/>^ = (1000-A)/1000. 



' This is the celebrated Laplacian generalization of Bayes' formula. No attempt 

 to demonstrate it will be made here. The subject is dealt with at length by Laplace 

 in the Theorie .Analytique des Probabilites and by Poisson in the Recherches Sur 

 L:i Probabilite des Jugemcnts. .A beautiful and relatively short demonstration 

 is given by Poincare in his Calcul des Probabilites. 



