Page:Philosophical Transactions - Volume 053.djvu/484

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a direction to our judgment that may be of considerable use till some person slall discover a better approximation to the value of the two series's in the first rule[1].

But what most of all recommends the solution in this Essay is, that it is compleat in those cases where information is most wanted, and where Mr. De Moivre's solution of the inverse problem can give little or no direction; I mean, in all cases where either p or q are of no considerable magnitude. In other cases, or when both p and q are very considerable, it is not difficult to perceive the truth of what has been here demonstrated, or that there is reason to believe in general that the chances for the happening of an event are to the chances for its failure in the same ratio with that of p to q. But we shall be greatly deceived if we judge in this manner when either p or q are small. And tho' in such cases the Data are not sufficient to discover the exact probability of an event, yet it is very agreeable to be able to find the limits between which it is reasonable to think it must lie, and also to be able to determine the precise degree of assent which is due to any conclusions or assertions relating to them.

  1. ↑ Since this was written I have found out a method of considerably improving the approximation in the 2d and 3d rules by demonstrating that the expression comes almost as near to the true value wanted as there is reason to desire, only always somewhat less. It seems necessary to hint this here; though the proof of it cannot be given.

LIII. An

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