XL1X j litti'inlurtiou xxix 



Now l)i Muivn neglects terms of the order t or higher orders an compared 



II 



with - . This enables him to write 



P(n, 1, t')= l - 



Example. In an army corps of 10,000 men kept up to full strength, what is the 

 number that must be lost before the odds are equal that all the original members 

 of the corps have been replaced ? 



Here P(n, l,i) = , an<l n= 10,000. 



,, / 9999 ) w . 



lhus 



= ! 2 10 ' 000 -log(2 10 ' 000 -l) 

 ~log 10,000 - log 9999 



But jQ log 2 = -00003,01030, 



2 io,ooo = 1-00006,93181, 

 i 

 2W, -! = -00006,93181, 



i 

 log (2 10 - *- 1) = 5-84034,66502 = - 4'15915,33498 ; 



log 9999 = 3-99995,65684. 



. _ -00003,01030 + 4-15915,334!).s 

 ~4^3-99995,65684 



_ 4-15918,34528 _ 

 - -00004,34316 ' 



Laplace * obtains for the same problem differently worded, and by processes of 

 approximation which give little opportunity of judging the order of approximation, 

 the answer 



95767-4. 



Assuming this agreement shows that both are practically correct, De Moivre's 

 approximation involves far less arithmetic, and fewer ill-defined approximations than 

 Laplace's formula f, and should accordingly be adopted for practical statistical work. 



* Loc. cit. pp. 193 200. 



t I have reworked Laplace's formula and find i to be 



.................. (vi), 



where T is the desired value of P (n, 1, i). 



k [Continued on next page 



