ELEMENTARY SAMPLING THEORY FOR ENGINEERING 



29 



It may be helpful at this point to state and solve a simple problem, 

 which will serve to bring out the fundamental principles involved. 

 An urn is known to contain 10 balls, some of which are white and the 

 others black. Five balls have been drawn and not replaced. Of 

 these five, one is white and four are black. What is now the proba- 

 bility that the urn originally contained just one white ball and nine 

 black? Two white and eight black? 



Before we proceed to obtain a solution for this problem we have 

 to make some assumption, based on knowledge available before the 

 drawings were made, concerning the probability that the urn contains 

 black and white balls in any given proportion. 



Consider two such assumptions — 



(a) All proportions are a priori equally likely, i.e., before the 

 drawings it is as likely that three whites and seven blacks were put 

 in the urn as six whites and four blacks, etc. 



{h) The urn was filled with ten balls drawn at random from a bag 

 containing a very large number of balls of which a quarter are white 

 and the remainder are black. 



There are, before the drawings, 11 possible hypotheses concerning 

 the contents of the urn. They range from whites and 10 blacks 

 to 10 whites and blacks, as listed in the two left-hand columns of 

 Table I and shown in Fig. 1. The probability in favor of each of 



0.3 



0.2 

 0.1 

 0.0 



0.4 



0.2 



0.0 



8 



10 



Fig. 1. The upper curve shows two different assumptions concerning the 

 a priori probabilities, while the lower pair shows the a posteriori probabilities. In 

 both cases the dots refer to the hypothesis of uniform a priori probability while the 

 circles refer to the assumption that the urn itself is a random sample from a large 

 stock of which one fourth of the balls are white. 



these hypotheses is the "a priori existence probability" in favor of 

 the hypotheses, and is represented by the symbol w(X), X referring 

 to the number of white balls assumed to be in the urn. 



