542 



THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1957 



of n and for d = d* we .shall be particular!}^ interested in the value p(^] = 

 P[i] (d*;n) since this (as shown in Appendix II) gives the smallest 

 probability Pes of a correct selection for all the configurations included 

 in the statement of the experimenter's specification. This particular con- 

 figuration (6) with d = d* and p^] = pl^ (which depends on n) is 

 called the Least-Favorable Configuration. 



Although the least fa\-orable configuration depends on n, it has been 

 empirically found that for n ^ 10 (and in some cases for n ^ 4) the least 

 favorable configuration is approximately given by p[i] = I (1 + d*) 

 in which the two values, p[i] and p^2] = Pm — d* are symmetric about §, 

 This symmetric configuration clearly does not depend on n. Fig. 1(b) 

 shows that as n —^ y^ the least favorable configuration approaches this 

 symmetric configuration (i.e., the straight line marked n = x) q^(ite 

 rapidly for any value of d. In Appendix III it is proved that the sym- 

 metric configuration is least favorable as n -^ x. Fig. 1(a) shows for 

 k = 3,n = 10, and any vsdue of d the error in Pes which arises as a result 

 of using the symmetric configuration instead of the true least favorable 

 configuration. 



0.0002 



a. 

 o 



(T 0.0001 

 a. 



LU 





 1.0 



0.9 



or O.J 



cr 

 o 



0.7 



0.6 



0.5 



0.1 0.2 0.3 0.4 0.5 0.6 



d (or d*) 



0.7 



0.8 



0.9 



1.0 



Fig. 1 — (a) Error in P cs as a result of using the sj-mmetric configuration in- 

 stead of the least favorable configuration for k = 3, n = 10, and any common true 

 difference d. (b) Least favorable value pd] (d) of pd) as a function of the common 

 true difference d = p[i] — pn] , i ^ 2, for A- = 3 and selected values of n. (for 



d = d*, Pdi (d) = pui) 



