1/2, the truth being 1/3. 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—
half the time within 0.477([sqrt]((2p(1-p))/s)
9 times out of 10 within 1.163([sqrt]((2p(1-p))/s)
99 times out of 100 within 1.821([sqrt]((2p(1-p))/s)
999 times out of 1,000 within 2.328([sqrt]((2p(1-p))/s)
9,999 times out of 10,000 within 2.751([sqrt]((2p(1-p))/s)
9,999,999,999 times out of 10,000,000,000 within 4.77([sqrt]((2p(1-p))/s)
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 1/2; hence 2p(1-p) is also 1/2. The number of white children counted was near 1,000,000; hence the fraction whose square-root is to be taken is about 1/2000000. The root is about 1/1400, and this multiplied by 0.477 gives about 0.0003 as the probable error in the ratio
