Page:Chance, love, and logic - philosophical essays (IA chancelovelogicp00peir 0).pdf/143

 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 366/1000; that of drawing one white ball would be 370/1000; that of drawing two would be 185/1000; that of drawing three would be 61/1000; that of drawing four would be 15/1000; that of drawing five would be only 3/1000, 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:

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 99 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