Page:Popular Science Monthly Volume 12.djvu/735

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ILLUSTRATIONS OF THE LOGIC OF SCIENCE.

is this: Given a certain state of things, required to know what proportion 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 relative 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 contrary, 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 represent 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 14, and 24 times out of 81 we should find it 12, the truth being 13. 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{\tfrac{2p(1-p)}{s}}
9 times out of 10 within 1.163 \sqrt{\tfrac{2p(1-p)}{s}}
99 times out of 100 within 1.821 \sqrt{\tfrac{2p(1-p)}{s}}
999 times out of 1,000 within 2.328 \sqrt{\tfrac{2p(1-p)}{s}}
9,999 times out of 10,000 within 2.751 \sqrt{\tfrac{2p(1-p)}{s}}
9,999,999,999 times out of 10,000,000,000 within 4.77 \sqrt{\tfrac{2p(1-p)}{s}}