ILLUSTRATIONS OF THE LOGIC OF SCIENCE. 715 



is this : Given a certain state of things, required to know what pro- 

 portion of all synthetic inferences relating to it will be true within a 

 given degree of approximation. Now, there is no difficulty about this 

 problem (except for its mathematical complication) ; it has been much 

 studied, and the answer is perfectly well known. And is not this, 

 after all, what we want to know much rather than the other ? Why 

 should we want to know the probability that the fact will accord with 

 our conclusion ? That implies that we are interested in all possible 

 worlds, and not merely the one in which we find ourselves placed. 

 Why is it not much more to the purpose to know the probability that 

 our conclusion will accord with the fact ? One of these questions is 

 the first above stated and the other the second, and I ask the reader 

 whether, if people, instead of using the word probability without any 

 clear apprehension of their own meaning, had always spoken of rela- 

 tive frequency, they could have failed to see that what they wanted 

 was not to follow along the synthetic procedure with an analytic one, 

 in order to find the probability of the conclusion ; but, on the con- 

 trary, to begin with the fact at which the synthetic inference aims, 

 and follow back to the facts it uses for premises in order to see the 

 probability of their being such as will yield the truth. 



Aa we cannot have an urn with an infinite number of balls to rep- 

 resent the inexhaustibleness of Nature, let us suppose one with a finite 

 number, each ball being thrown back into the urn after being drawn 

 out, so that there is no exhaustion of them. Suppose one ball out of 

 three is white and the rest black, and that four balls are drawn. Then 

 the table on page 713 represents the relative frequency of the different 

 ways in which these balls might be drawn. It will be seen that if we 

 should judge by these four balls of the proportion in the urn, 32 times 

 out of 81 we should find it , and 24 times out of 81 we should find it 

 -J, the truth being ^. To extend this table to high numbers would be 

 great labor, but the mathematicians have found some ingenious ways 

 of reckoning what the numbers would be. It is found that, if the true 

 proportion of white balls is p, and s balls are drawn, then the error of 

 the proportion obtained by the induction will be 



/ 2p(l-p ) 

 /2p{l-p) 



s 

 3( 1 -/>) 



half the time within 0.477 



9 times out of 10 within 1.163 



99 times out of 100 within 1.821 



/2(1 p) 

 999 times out of 1,000 within 2.328 y 



/2p(l-p) 

 9,999 times out of 10,000 within 2.751 y 



/2p(lp) 

 9,999,999,999 times out of 10,000,000,000 within 4.77 y 



