60 



ESSAY ON PROBABILITIES. 



ing gives a white ball, and the ball is replaced. What 

 is the chance that a second drawing shall give a black 

 ball ? 



The preceding states under which the first event may 

 have happened, are 



(2 white) (1 white, 1 black); 



and 1 and % are the chances of a white ball, if one or 

 other state were absolutely known to exist. Hence., by 

 the last principle, -| and -^ are the chances which the 

 observed e*vent gives to the two states; that is, it is 

 two to one that both balls were white. Now, the black 

 ball can only appear at the second drawing, upon the 

 supposition of the second state existing ; and this sup- 

 position being made, the chance of a black ball at the 

 second drawing is -J. Hence, page 43., the second 

 event depending upon two contingencies, of which the 

 chances are ^ and -J, its chance is J, or it is five to 

 one against the second drawing being black. But let 

 us now ask what is the chance of a white ball at the 

 second drawing ? Either of the preceding states admit 

 of such an event, and, in fact, the event proposed a 

 white ball at the second drawing means 



One or other of these C (2 white) and white drawn. 



two combinations ( (1 white, 1 black) and white drawn. 



In the first combination, the first contingency (the 

 chance of which is -J) ensures the second: so that 

 A- x 1 is the chance of a white ball being drawn, and 

 of (2 white) being the lottery from which it is drawn. 

 In the second combination, the chances of the two con- 

 tingencies are -] and 1, whence J- is the chance of a 

 white ball being drawn, and being drawn from (1 

 white, 1 black). But the event proposed happens if 

 either of these cases occur ; therefore, - 4. l, or | , is 

 the chance of a white ball at the second drawing, as 

 might have been inferred from the probability already 

 obtained for a black ball. By such reasoning as the 

 preceding, the following principle is established : 



Principle VI. Having given an observed event A, 



