94 BEI.I. SYSTEM TECHNICAL JOURNAL 



We iiia\-, therefore, rewrite (1) as follows: 



w'{pk)B'{i,\m, Pk) .„. 



^W'iPs)B' {7,100, Ps) 



s = l 



wiiere It". IV are the forms assuiiied liy the functions \V, B, respec- 

 tively, when the ratio p<. is used instead of the number K. 

 The interpretation of the terms of the expansion of the binomial 



(p + g)""' tells us that 



B'{7,\()0,p)= 0^)p'{\-py'=(^^)pV 



where ( - ) i^ 'I symbol for the number of combiii.itions ol lOO 



things 7 at a time. 



Substituting in (2) and canceling from numerator and denominator 



the conunon lactor I - ) gnes 



p w'(p,)p,m-p,r' . 



2 ]v'(ps)ps'n-p.)'-" 



1 



I""rom ['A) we obtain tor (lie a posteriori |)rob,ibilil\- thai the ratio ol 

 white balls does nol exceed A'; 1 ,()()(), 



/'(A->A'.,) = 2A-. 

 1 



Likewise, the a posteriori probabilit\- that the ratit) is not less than 

 A' i/ 1,000 is 



P[K<K,) = ^Pk. 



Finally, the a [josteriori probai>ilit\ llial the lalio is not less than 

 A',/1,000 or greater than A, 1, ()()() is 



K, %w'{p.)p,'{i-p,r' 



/^(A-,<A'^A%) = 2/'*=|5 W 



"' '^n-'(p^p,-(\-Psy' 



