The use of this may be illustrated by an example. By the census 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 difference 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 circumstances? Here p may be taken at 12; hence 2p (1-p) is also 12. The number of white children counted was near 1,000,000; hence the fraction whose square-root is to be taken is about 12000000. The root is about 11400, 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 censuses, 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 drawings. The probability of drawing no white ball would be 3661000; that of drawing one white ball would be 3701000; that of drawing two would be 1851000; that of drawing three would be 611000; that of drawing four would be 151000; that of drawing five would be only 31000 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 cannot, determine the probability of synthetic inference. When I reason in this way:
But Epimenides is a Cretan;
I know that reasoning similar to that would carry truth 99 times in 100. But when I reason in the opposite direction:
But these were all atrocious 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 proportion 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 16. So much
