ccxxxviii Tables for Statisticians and Biometricians [XLIX 



Illustration (ii). If u = 1, we see that the chance that in i drawings or prickings 

 all individuals of a group of n will have been drawn or marked is 



(iv). 



n 



Example. Five platoons of ten men are kept each at full strength, but each 

 loses 10/ in every engagement. In how many engagements must they be con- 

 cerned before the chances are about 7 to 3 that one or more of the platoons 

 contains none of the original members ? 



Consider one platoon, the chance that after 19 engagements none of the original 

 men will be left equals 



A 10 19 19 ' 

 POO, 1,19) = = ,7 (10, 19) 



= -012,1645 x 14-238,2675 

 = -17320. 



This indicates that about one platoon in six would be denuded of its original men. 

 We next try 20 engagements and find 



A10A20 90 t 



P(10, 1,20) = -^ r - ^0(10, 20) 



= -024,3290 x 8-826,386 

 = -214,737. 

 The chance therefore that a single platoon will not have lost all its original 



members is 



785,263, 



or the chance that out of five platoons none will have lost all its original members is 



(785,263) 5 = -298,590. 

 Hence the chance that one or more will have lost all its original members is 



701,410, 

 or the odds are about 7 to 3. 



Illustration (iii). In questions of the above kind we may meet with problems 

 which involve higher values of n and i than are provided for in our Table. A some- 

 what lengthy series of approximations with a complicated formula is given by 

 Laplace*, but provides in the case of high values of n, no better result than a very 

 simple formula of De Moivref. We have 

 P(n, 1, i)= A B OV* 



* Loc. cit. p. 200. 



t The Doctrine of Chances, 3rd Edition, 1756, pp. 123126, and Preface, pp. ix x. De Moivre, 

 following a suggestion of Halley, replaces an arithmetical by a geometrical series, which here seems 



/ 1\ 8 s 



equivalent to taking II J =1 . 



