276 BELL SYSTEM TECHNICAL JOURNAL 



Now regarding the general a posteriori prol)lem, 



"An event has liai)i)ened which must liave arisen from some one of a given number 

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



what do the two examples we have just considered suggest? In both 

 problems we inquired into: 



1. The frequency with which each of the possible causes is met with 



BEFORE THE OBSERVED EVENT HAPPENED. This frequency 



is called the a priori existence probability for the corresponding 

 cause. 



2. The probability that a cause, if brought into play, would reproduce 



the observed event. This probability will hereafter be referred 

 to as the a priori productive probability for the cause in question. 



In the case of the broken ankle, the a priori existence probabilities were 

 equal and took no part in our conclusion; we based the a posteriori 

 probabilities entirely on the a priori productive probabilities. We did 

 just the opposite with reference to the coin spun by the Cleveland 

 financier; on account of the equality of the a priori productive proba- 

 bilities we deduced a posteriori probabilities in terms of the unequal a 

 priori existence probabilities. 



It is apparent that our two examples represent extreme cases. In 

 general, the solution of an inverse or a posteriori problem, involving a 

 number of causes, one of which must have brought about a certain 

 observed event, depends on both sets of direct, or a priori probabilities. 

 Those of the first set give the frequency with which the various causes 

 were to be expected before the observed result occurred ; those of the 

 second set give the frequencies with which the observed result would 

 follow from the various causes if each were brought into play. 



II 



Bearing in mind the two distinctly different sets of a priori proba- 

 bilities required in arriving at a posteriori conclusions regarding the 

 possible causes of an observed event, we must now give some thought 

 to the algebra of the subject before taking up Bayes' problem and 

 theorem. For this purpose consider the following bag problem: 



A bag contained M balls of which an unknown number were white. 

 From this bag N balls were drawn and of these T turned out to be 

 white. What light does this outcome of the drawings throw on the 

 unknown ratio of the number of white balls to the total number of 

 balls, M, in the bag? Let x be this unknown ratio. 



Two cases of this problem may be considered: 



