58 ESSAY ON PROBABILITIES. 



1. That the black ball is with the three white ones. 



2. That the first drawing is from the lottery which has 

 the black ball, and the second from the other, is (case 7) 

 -J-g-f- . To find the total probability that the black ball 

 is with the three white ones, we must add the proba- 

 bilities of all the cases (as to drawings) which can take 

 place under this arrangement, namely -f-^ y -J-J-^ -|Jv 

 J-4T> g ivin S iti- Consequently, from the observed 

 event, it is slightly more probable that the black ball is 

 with the three white ones, than with the two. 



The principle which we have illustrated, though a 

 mathematical consequence of those which precede, is 

 nevertheless received in common life upon its own 

 evidence. When an event happens, we immediately 

 look to that cause or antecedent which such event most 

 often follows. When it rains, we suspect the barometer 

 must have fallen ; because, when the barometer falls, 

 it usually rains. 



Our next step is to inquire, what is the probability 

 which an event gives to its several possible antece- 

 dents, upon the supposition that they are not all equally 

 likely beforehand ; as in the following instance. 



PROBLEM. A white ball is drawn, and from one or 

 other of the following urns : 



(3 white, 4 black) (2 white, 7 black) : 

 but before the drawing was made, it was three to one 

 that the drawer should go to the first urn, and not to 

 the second. What is the chance that it was the first 

 urn from which the drawing was made ? 



We may immediately reduce the preceding to the 

 case where all the antecedent circumstances are equally 

 probable, by introducing urns enough of the first kind 

 to make it 3 to 1 that the drawing is made from one 

 or other of them. Let us suppose the urns to be as 

 follows : 



(3 white, 4 black) (3 white, 4 black) (3 white, 4 black) 

 (2 white, 7 black): 



these urns being equally probable, the hypothesis of 



