Page:Philosophical Transactions - Volume 053.djvu/484

 a direction to our judgment that may be of conideruble ue till ome peron hall dicover a better approximation to the value of the two eries's in the firt rule.

But what moft of all recommends the olution in this Eay is, that it is compleat in thoe caes where information is mot wanted, and where Mr. De Moivre’s olution of the invere problem can give little or no direction; I mean, in all caes where either p or q are of no considerable magnitude. In other caes, or when both p and q are very coniderable, it is not difficult to perceive the truth of what has been here demonstrated, or that there is reaon to believe in general that the chances for the happening of an event are to the chances for its failure in the ame 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 mall. And tho’ in uch caes the Data are not Sufficient to dicover the exact probability of an event, yet it is very agreeable to be able to find the limits between which it is reaonable to think it mut lie, and alo to be able to determine the precie degree of aent which is due to any concluions or aertions relating to them.

f Since this was written I have found out a method of coniderably improving the approximation in the 2d and 3d rules by demontrating that the expreion 1 + 2 E a? $ + 2 Ea? In comes n almot as near to the true value wanted as there is reafon to deire, only always omewhat les. It eems neceary to hint this herej though the proof of it cannot be given. LIII. An