92 BELL SYSTEM TECHNICAL JOURNAL 



In most problems of this type the determination of the productive 

 probabihty for each hypothesis is a question of pure mathematics. 

 But when we proceed to evaluate the a priori existence probability 

 for each hypothesis or cause, common sense and guessing must fre- 

 quently be resorted to. The history of the applications of a posteriori 

 probability is so full of parado.xes resulting from appeals to common 

 sense that to some high authorities the whole theory is a fallacy. 

 Prof. George Chrystal ^ closes a severe attack on Laplace's Theorie 

 Anal>'tique with the statement — "The indiscretions of great men 

 should be quietly allowed to be forgotten." Nevertheless, the writers 

 will assume the Laplacian view of the subject, especially as it has been 

 defended by such authorities as Karl Pearson and E. T. Whittaker. 



The above typical problem has been introduced because its mere 

 statement leads us immediately to the conceptions of existence and 

 producti\e probabilities with reference to different possible hypotheses. 

 But, it is not our intention to bring any notoriety on the young man 

 by answering the questions raised. Moreover, the hypotheses made, 

 differ qualitatively, whereas, our telephone problem inv'olves various 

 hypotheses which differ only quantitatively. We, therefore, proceed 

 to another typical problem, a solution of which will give us at once 

 the solution of the telephone problem. 



A bag contains 1,000 balls; an unknown number of these are white 

 and the rest not white. Of 100 balls drawn 7 are found to be white. 

 What light does this information throw on the value of the unknown 

 number of white balls? What is the probability that there are 70 

 white? Is it a s;ife Ix-t that the number of white balls lies lictween 

 60 and 80? 



Two cases of tliis problem may be considered: 



Case 1. After a ball is drawn it is replaced and the bag is shaken 



thoroughly before the next drawing is made. 

 Case 2. A drawn liall is not replaced before another ball is drawn. 



These two cases become essentially identical if the total number 

 of balls in the bag is very large compared with the number drawn.' 

 In the following discussion Case 1 is assumed. 



The information at hand is that 100 drawings resulted in 7 whites. 

 Obviously the bag contains at least one white, but we are free to 

 choose between 1M)9 possible hypotheses. 



• TraiiSiictions of the Acturial Society of Edinburgh — Vol. II, No. 13 — On Some 

 Fundamental Principles in the Theory of Probabilities. 



' ¥oT the application to practice herein contemplated it is thouglit that the number 

 of balls in the bag should be at least ten times the number drawn. 



