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BELL SYSTEM TECHNICAL JOURNAL 



Under assumption "a" (Case 1) each hypothesis has a probability 

 of 1/11 or .090909. Under assumption "6" (Case 2) the probability 

 that the urn contains X whites and 10 — X blacks is the binomial 



TABLE I 



In the column headed px we give the productive probabilities for 

 both cases. These are the probabilities that five drawings from an urn 

 whose contents were as given by the corresponding hypothesis would 

 yield the observed one white ball and four black. These are zero in 

 the case of X = 0, 7, 8, 9 and 10 since urns so constituted could not 

 have given the observed drawings. 



For the other cases, the productive probability is the ratio 



10 - X 



4 



10 



5 



In this expression the denominator is the total number of combi- 

 nations of 10 balls taken five at a time, and the numerator is the 

 number of ways of selecting one out of X white balls and four out 

 of the remaining 10 — X black balls. These figures are tabulated in 

 Table I under the heading px- 



We now have all of the component parts of our problem under the 

 two different assumptions "a" and "6." It only remains to apply 

 "Bayes' Rule."- Now the generalized Bayes Rule tells us that the 

 a posteriori probability, Px, in favor of an hypothesis ajier the drawings 



2 This rule was first enunciated by an English cleric, Bayes by name, in a memoir 

 in Philosophical Transactions for 1763. It was generalized by Laplace in 1812 to 

 cover cases not etjually likely. 



