278 



BELL SYSTEM TECHNICAL JOURNAL 



taken T at a time. Substituting in (1) we obtain, after canceling 

 from numerator and denominator the common factor ( 7^ j . 





Pk = 



(2) 



If in eciuation (2) we give k successively the values a, a + 1, a + 2, 

 • • • h — \, b and add the results we have 



Pa + Pa + 1 + 



+ Pt 



or 



P{Xa, Xb) 



L w{Xk)Xk''X^ - Xk)-^'' 



£; w{xk)xk'^{i - Xk)""' '^ 



A=0 



(3) 



for the a posteriori probability that the unknown ratio of white to 

 total balls in the bag lies between a/AI and b/M; both inclusive. 



Ill 

 Bayes' Problem 



Consider the table represented by the rectangle A BCD in Fig. 1. 

 On this table a line OS was drawn parallel to, but at an unknown 

 distance from, the edges AD and BC. Then a ball was rolled on the 

 table N times in succession from the edge AD toward the edge BC. 

 As indicated in the figure, it was noted that T times the ball stopped 

 rolling to the right of the line OS and .V - T times to the left of that 



line. 



What light does this information shed on the unknown distance 

 from ^L> to OS? In more technical terms, what is the a posteriori 

 probability that the unknown position of the line OS lies between any 

 two positions in which we may be interested? 



C 



S 



B 



1 



O 



D 



O 



A 



I'ig. 1. 



