X I j I X I /ntrod'uction cc\ \ \ N i i 



The i|ii;int.it.y t:il lcd is the function 



\vln-n- 



Illustration (i). Given a list of tho names of // individuals, if a of these be 

 pricked at random i times, what is the chance that all the names will have Ix-cn 

 pricked ? 



The answer to this problem in the language of the lottery is given by Laphu-r*. 



............ (Hi), 



where t is to be put equal to zero after the differencing. 



It is clear that by expanding in powers of t and then putting t zero, the 

 required probability will be expressed in terms of the differences of zero. 



For example, on a court of discipline of four members two are taken at random 

 for a given sitting; what is the chance that all four will have served in six sittings? 



P(4, 2, 6)=A 4 ((*-1)) 6 /12 6 , 

 or with t put zero after differencing: 



= (A 4 12 - 6A 4 O n + 15A 4 10 - 20A 4 9 + 15A 4 8 - 6A 4 7 + A 4 0)/12 



p x -030,638,779 - - t |? x -087,632,275 



\- * fc. ' r 



zr X -225,562,169 - x -513,888,889 



I7o ^^ 



+ - Q ~ x 1-012,500,000 -* ^x l-666,666,6(7 

 4 



2-166,666,667, 



from the values of q (p, s) in Table XLIX, paying regard to the factor F (p + 8 + 1). 

 Hence P (4, 2, 6) = 4-9149,7080 - 7-0288,3872 



+ 4-1118,1037 - 1-2490,3549 

 + -2050,7813- -0168,7886 

 + -0005,2244 



9-2323,8174- 8-2947,5307 

 = -9376,2867. 



Thus in about 100 trials there would be only six cases of all four members failing 

 to serve at least once in sets of six sittings. 



* Thlorie analytique des probability*, 1st Edn. 1812, p. 193, or, in any edition, Livrc n. chap. ii. 



