THE STATE OF THE ODDS. 305 



added to three) are white. There remain seven black 

 balls, and therefore the odds are 8 to 7 on the pair. 



To impress the method of treating such cases on the 

 mind of the reader, we take the betting about three 

 horses say 3 to 1, 7 to 2, and 9 to 1, against the three 

 horses respectively. Then their respective chances 

 are equal to the chance of drawing (1) one white ball 

 out of four, one only of which is white; (2) a white 

 ball out of nine, of which two only are white ; and (3) 

 one white ball out of ten, one only of which is white. 

 The least number which contains four, nine, and ten, 

 is 180; and the above chances, modified according to 

 the principle explained above, become equal to the 

 chance of drawing a white ball out of a bag containing 

 180 balls, when 45, 40, and 18 (respectively) are white. 

 Therefore, the chance that one of the three will win is 

 equal to that of drawing a white ball out of a bag con- 

 taining 180 balls, of which 103 (the sum of 45, 40, and 

 18) are white. Therefore, the odds are 103 to 77 on 

 the three. 



One does not hear in practice of such odds as 103 

 to 77. But betting-men (whether or not they apply 

 just principles of computation to such questions, is 

 unknown to us) manage to run very near the truth. 

 For instance, in such a case as the above, the odds on 

 the three would probably be given as 4 to 3 that is, 

 instead of 103 to 77, or, which is the same thing. 412 

 to 308, the published odds would be 412 to 309. 



