ON INVERSE PROBABILITIES. 53 



CHAPTER III. 



ON INVERSE PROBABILITIES. 



IN the preceding chapter, we have calculated the 

 chances of an event, knowing the circumstances under 

 which it is to happen or fail. We are now to place 

 ourselves in an inverted position : we know the event, 

 and ask what is the probability which results from the 

 event in favour of any set of circumstances under which 

 the same might have happened. This problem is fre- 

 quently enunciated as follows: An event has hap- 

 pened, such as might have arisen from different causes : 

 what is the probability that any one specified cause did 

 produce the event, to the exclusion of the other causes ? 

 By a cause, is to be understood simply a state of things 

 antecedent to the happening of an event, without the 

 introduction of any notion of agency, physical or 

 moral. 



In order that we may secure a problem of sufficient 

 simplicity, we must limit the number of possible ante- 

 cedent states. Let us suppose that there is an urn, of 

 which we know that it contains balls, three in number, 

 and either white or black, all cases being equally pro- 

 bable : that is, before any drawing takes place, all we 

 can say is, that we are going to draw out of one of the 

 following, having no reason for supposing one in pre- 

 ference to another 



A B C DEF 



!.() II. ( o ) III. ( o o ) IV. ( o o o) 



A drawing takes place, and a white ball is produced, 

 consequently I. is immediately excluded ; for from it 

 the observed event could not have been produced. This 

 much is certain ; but we are also tempted to say that 

 E 3 



