O* ESSAY ON PROBABILITIES. 



II. is rendered unlikely, because, from such an ante- 

 cedent state of things, a black ball would have been 

 more likely than a white one. On the same prin- 

 ciple III. is more likely than II., and IV. the most 

 likely of all. We have then to decide the relative pro- 

 babilities of II., III., and IV. 



Before the drawing took place, the probability of each 

 set of circumstances was J ; and, the lottery being 

 given, the probability of any one ball in it was i. 

 Thus the chance of III. being the lottery, and the 

 second white ball being drawn from it, was x -^, or 

 -^ . The same of other balls : so that, in fact, our 

 primitive position was that of having to draw from 12 

 balls, 6 white and 6 black, all equally probable. But 

 the observed event changed that position ; a white ball 

 was drawn : was it a given ball (namely, the white ball 

 in II.), or was it one of two given balls (those in III.), 

 or was it one of three (those in IV.) ? There are six 

 cases in question, namely, A, B, C, D, E, F, and one 

 of them happened, we do not know which. We have 

 used all the knowledge we have (namely, that a white 

 ball was drawn,) in excluding the black balls. 



Hence the chance that A was drawn, or that 



II. was the lottery - is -J- : 



That either B or C was drawn, or that III. 



was the lottery - - is |- : 



That either D, E, or F was drawn, or that 



IV. was the lottery - - is -|-. 



In the preceding instance, owing to the number of balls 

 being the same in every lottery, the antecedent proba- 

 bility of each ball was the same. Previous to deducing 

 a rule, I take an instance in which this is not the case. 



PROBLEM. A white ball has been drawn, and from 

 one or other of the two following urns : 



(2 white, 5 black) (3 white, 1 black). 

 What are the probabilities in favour of each urn ? 



The case is not now that of a lottery of 5 white and 

 6 black balls ; for the chance of our going to the first 

 urn (which is-l), and thence drawing a given white 



