CALCULATION OF CHANCES. 73 



by experience, of the proportion between the cases in 

 which the fact occurs, and those in which it does not 

 occur. Every calculation of chances is grounded on 

 an induction : and to render the calculation legitimate, 

 the induction must be a valid one. It is not less an 

 induction, though it does not prove that the event 

 occurs in all cases of a given description, but only that 

 out of a given number of such cases, it occurs in about 

 so many. The fraction which mathematicians use 

 to designate the probability of an event, is the ratio of 

 these two numbers ; the ascertained proportion be- 

 tween the number of cases in which the event occurs, 

 and the sum of all the cases, those in which it occurs 

 and in which it does not occur taken together. In 

 playing at cross and pile, the description of cases 

 concerned are throws, and the probability of cross is 

 one half, because it is found that if we throw often 

 enough, cross is thrown about once in every two 

 throws ; and because this induction is made under 

 circumstances justifying the belief that the proportion 

 will be the same in other cases as in the cases 

 examined. In the cast of a die, the probability of ace 

 is one-sixth ; not, as Laplace would say, because there 

 are six possible throws, of which ace is one, and 

 because we do not know any reason why one should 

 turn up rather than another ; but because we do know 

 that in a hundred, or a million of throws, ace will be 

 thrown about one-sixth of that number, or once in 

 six times. 



Not only is this third condition indispensable, but 

 if we have that, we do not want Laplace's two. It is 

 not necessary that we should know how many possi- 

 bilities there are, or that we should have no more rea- 

 son for expecting one of them than another. If a north 

 wind blows one day in every ten, the probability of a 



