Page:Mind (New Series) Volume 9.djvu/97

SYMBOLIC REASONING. 83 data will neither increase nor diminish the chance of the other being true. A B Putting a for (the chance that A is true) and b for - (the chance that B is true), we have the formulae A a B j s A a S B The second of these shows that (since a and b are necessarily A B positive) 8^ and S-j- have always the same sign ; that is, they are either both positive, or both negative, or both zero. Whether A and B be independent or not, we have the formula which we have called , namely, AB A B B A B , A T = 7" A = 7-B = "A = 6 B" Hence, when A aiid B are independent, we get AB A B . = . = ao ; f which expresses another well-known truth in probability. T> T> For the supposition of independence implies that -v = . To take a more serious subject of illustration. Let S assert that a person now forty, and taken at random out of all the persons of that age now in England, will reach the age of seventy ; and let A assert that his occupation is A. a The chance may be found approximately from tables of g statistics ; and we will suppose that the chance -.- may be A. g ascertained in a similar manner. In that case 8-^-, the A. dependence of S upon A, will be known and may be taken as an approximate measure of the healthiness (when positive) or unhealthiness (when negative) of the occupation A. A S S S statistical series 8-^, 8^, 8p, etc., might thus be convenient A. -D v> for affording comparisons of the healthiness or unhealthiness of different occupations during that period of life. As an illustration of the working of this notation, I give a proof of the formula A _ A _ 1 A B B' F B'