Page:Philosophical Transactions - Volume 053.djvu/484

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a direction to our judgment that may be of conſideruble uſe till ſome perſon ſhall diſcover a better approximation to the value of the two ſeries's in the firſt rule.

But what mofſt 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.

f 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 1 + 2 E a? $ + 2 Ea? In comes n almoſt as near to the true value wanted as there is reafon to deſire, only always ſomewhat leſs. It ſeems neceſſary to hint this herej though the proof of it cannot be given. LIII. An