2; 8 LIGHT SCIENCE FOR LEISURE HOURS. 



white ball out of a bag of three, one only of which is 

 white. In like manner, the chance of the second horse 

 is equal to that of drawing one white ball out of a bag 

 of five, one only of which is white. Now we have to 

 find a number which is a multiple of both the numbers 

 three and five. Fifteen is such a number. The chance 

 of the first horse, modified according to the principle 

 explained above, is equal to that of drawing a white 

 ball out of a bag of fifteen of which five are white. In 

 like manner, the chance of the second is equal to that 

 of drawing a white ball out of a bag of fifteen of which 

 three are white. Therefore the chance that one of 

 the two will win is equal to that of drawing a white 

 ball out of a bag of fifteen balls of which eight 

 (five 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, let us 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. There- 



