ELEMENTARY SAMPLING THEORY FOR ENGINEERING 31 



have been made and taking account of the a priori information is 

 given by the ratio 



^ %v{X)px 



the summation in the denominator being extended over all possible 

 cases. 



The numerical values of this ratio are shown in the last two columns 

 of Table I, corresponding to the two assumptions in our problem 

 and also by the circles in Fig. 1. That we should have a different 

 set of results corresponding to the dififerent assumptions is to be 

 expected. It is interesting, however, that the difference in this case 

 is by no means great as Fig. 1 brings out. 



If after each drawing we had replaced the ball drawn, we would 

 have used for the productive probability px the binomial term 



. 5\/ XWIO - X 

 px 



1/Vio/ V 10 



since the successive drawings would not have changed the relative 

 constitution of the urn. The same would also be true if the urn 

 contained an indefinitely large number of balls with the same relative 

 proportions of black and white. 



Now if we agree that a white ball corresponds to a defective item 

 and a black ball to an acceptable item, we are immediately able, 

 by the use of these fundamental principles of a posteriori probability, 

 to write the general basic formal relation 



z »m(f)(rf) 



As we have just indicated, the troublesome element in this formula 

 is the function w{X) to which, in many practical problems, it is 

 difficult to assign any particular numerical values. In order to 

 proceed further, therefore, without detailed consideration of various 

 specific engineering problems we are forced to make some rather 

 general assumptions concerning the nature of the function w{X). 



Case I 



One of the most natural assumptions to make when no knowledge 



exists to the contrary is that zv{X) is a constant within that range 



^ It should be noted that in his original treatment of this formula Molina used 

 5 instead of 2 as the symbol for summation on account of the fact that finite inte- 

 gration entered into his analysis. Since in this presentation we are dealing only 

 with summation, we shall use the commoner form 2 to denote summation. 



