7 i 6 THE POPULAR SCIENCE MONTHLY. 



The use of this may be illustrated by an example. By the cen- 

 sus of 1870, it appears that the proportion of males among native 

 white children under one year old was 0.5082, while among colored 

 children of the same age the proportion was only 0.4977. The differ- 

 ence between these is 0.0105, or about one in a 100. Can this be 

 attributed to chance, or would the difference always exist among 

 a great number of white and colored children under like circum- 

 stances ? Here p may be taken at ; hence 2p (1p) is also $. The 

 number of white children counted was near 1,000,000 ; hence the 

 fraction whose square-root is to be taken is about aoooooo - The root 

 is about yj^, and this multiplied by 0.477 gives about 0.0003 as the 

 probable error in the ratio of males among the whites as obtained 

 from the induction. The number of black children was about 150,000, 

 which gives 0.0008 for the probable error. We see that the actual 

 discrepancy is ten times the sum of these, and such a result would 

 happen, according to our table, only once out of 10,000,000,000 cen- 

 suses, in the long run. 



It may be remarked that when the real value of the probability 

 sought inductively is either very large or very small, the reasoning is 

 more secure. Thus, suppose there were in reality one white ball in 

 100 in a certain urn, and we were to judge of the number by 100 draw- 

 ings. The probability of drawing no white ball would be ^\\\ ; that 

 of drawing one white ball would be y 3 ^ ; that of drawing two would 

 ^ e tWo ; tnat of drawing three would be yffg- ; that of drawing four 

 would be y^fo ; that of drawing five would be only T ^ i etc. Thus 

 we should be tolerably certain of not being in error by more than 

 one ball in 100. 



It appears, then, that in one sense we can, and in another we can- 

 not, determine the probability of synthetic inference. When I rea- 

 son in this way : 



Ninety-nine Cretans in a hundred are liars ; 



But Epimenides is a Cretan ; 



Therefore, Epimenides is a liar : 

 I know that reasoning similar to that would carry truth 09 times in 

 100. But when I reason in the opposite direction : 



Minos, Sarpedon, Rhadamanthus, Deucalion, and Epimenides, are 

 all the Cretans I can think of; 



But these were all atrocious liars, 



Therefore, pretty much all Cretans must have been liars ; 

 I do not in the least know how often such reasoning would carry me 

 right. On the other hand, what I do know is that some definite 

 proportion of Cretans must have been liars, and that this propor- 

 tion can be probably approximated to by an induction from five 

 or six instances. Even in the worst case for the probability of such 

 an inference, that in which about half the Cretans are liars, the ratio 

 so obtained would probably not be in error by more than . So much 



