. [ 418 ] 
a direction to our judgment that may be of confider- 
uble ufe till fome peri'on Shall difcover a better ap- 
proximation to the value of the two Series's in the 
£rft rule -J-. 
But what moft of all recommends the folution in 
this Effay is, that it is compleat in thole cafes where 
information is moft wanted, and where Mr. De 
Moivre’s folution of the inverfe problem can give 
little or no direction ; I mean, in all cafes where ei- 
ther p or q are of no considerable magnitude. In 
other cafes, or when both p and q are very confider- 
able, it is not difficult to perceive the truth of what 
has been here demonstrated, or that there is reafon to 
believe in general that the chances for the happening 
of an event are to the chances for its failure in the 
fame 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 fmall. And tho’ in fuch cafes the Data are not 
Sufficient to difcover the exadf probability of an event, 
yet it is very agreeable to be able to find the limits be- 
tween which it is reafonable to think it muft lie, and 
alfo to be able to determine the precife degree of affent 
which is due to any conclufions or alTertions relating 
to them. 
f Since this was written I have found out a method of confi- 
derably improving the approximation in the 2d and 3d rules by 
2 s 
demonftrating that the expreffion 1 + 2 E a? $ + 2 Ea? In comes 
n 
almoft as near to the true value wanted as there is reafon to defire, 
only always fomewhat lefs. It feems neceflary to hint this herej 
though the proof of it cannot be given. 
LIII. An 
