ESSAY ON PROBABILITIES. 



S 1 0,000100 = _ x 1 0,000*01 



101 



Consequently, the chance is TO T , or , nearly. 



-j-I-j- of 10,000106 106 



If the number of balls had been a million, instead 

 of 10,000, the preceding odds, namely, 101 to 5, 

 would still more nearly have represented the chance 

 that after 100 drawings, all white, the next five should 

 be white also. If the number of balls had been abso- 

 lutely unlimited, the preceding odds will correctly ex- 

 press that same chance. But a lottery with an unlimited 

 number of balls, each of which may be either white or 

 black, is a lottery which may be anything whatever. 

 For instance : [3 W, 7 B] is equivalent to an unlimited 

 lottery, in which for every 7 black balls there are 3 

 white ones. Again, an unlimited lottery in which any 

 number of balls may be black, and the rest white, is 

 one in which the chance of drawing a white ball may 

 be any whatever, and is absolutely unknown. The draw- 

 ing of a white ball from such a lottery may be likened 

 to the occurrence of an event, about the preceding 

 chances of which we are in total ignorance. The pre- 

 ceding process furnishes us with the following theorem. 

 When an event which may, for any thing we can see 

 to the contrary beforehand, happen in either of two 

 different ways, happens one way m times in succession, 

 it is m -f 1 to n that it shall happen n times more in 

 the same way, if it happen n times more at all. Thus, 

 suppose a person on the bank of a river, not knowing 

 in what country he is, and not having the smallest 

 reason to know whether the vessels which come up the 

 river carry flags or not : the first ten ships which 

 come up all carry flags (m = 10) ; then it is 10 -f 1 

 to 3, or 1 1 to 3 that the next three ships shall carry 

 them, and 10 4- 1 to 1, or 11 to 1 that the next ship 

 shall carry a flag. And it is always m -f 1 to 1, that 

 an event which has occurred in one out of two possible 

 ways m times in succession, shall happen the same way 

 on the next occasion. 



