274 BELL SYSTEM TECHNICAL JOURNAL 



I. Discuss briefly two problems each of which will emphasize one of 

 two kinds of a priori probabilities which should be constantly borne in 

 mind when Bayes' theorem is under consideration, 



II. Partially analyze a certain ball-drawing problem which will not 

 only serve as an introduction to the algebra of Bayes' theorem but will 

 later help to throw light on its significance, 



III. Present Bayes' problem and the related theorem, 



IV. Make some remarks on the value of the theorem and the contro- 

 versies which it raised. 



In carrying out this plan I shall find it convenient to ignore the 

 historic order of events. 



When probability is the subject under consideration one anticipates 

 problems such as: A coin is about to be tossed 15 times; what is the 

 probability that heads will turn up seven times? A sample of 100 

 screwdrivers is to be taken from a case containing 1000 screwdrivers of 

 which 300 are known to be defective; what is the probability that the 

 sample will contain 25 defectives? 



These are direct, or a priori, probability problems. In each of them 

 the nature of a game, or an experiment, is specified in advance and 

 then a question is asked relating to one, or more, of the possible out- 

 comes of the game or experiment. Problems of this type have occupied 

 the attention of mathematicians since the days of Pascal and Fermat, 

 the creators of the mathematical theory of probability. 



An inverse class of problems of great practical significance, called a 

 posteriori probability problems, came into prominence with the publi- 

 cation of Bayes' essay. In these we find specified the result or out- 

 come of a game which has been played, whereas the question then 

 asked is whether the game actually played was one or some other of 

 several possible games. This type of problems is usually stated as 

 follows : 



"An event has happened which nuist have arisen from some one of a given number 

 of causes: required the probability of the existence of each of the causes." 



I 



Consider this example: during his sophomore year Tom Smith 

 played on both the baseball and football varsity teams; we have been 

 informed that he broke his ankle in one of the games; what are the a 

 posteriori probabilities in favor of baseball and football, respectively, 

 as the baneful cause of the accident? 



Evidently the answer depends on the number of baseball and 

 football games played during their respective seasons and also on the 

 likelihood of a man breaking an ankle in one or the other of these two 

 games. As a concrete case assume that: 



