SELECTING THE BEST ONE OF SEVERAL BINOMIAL POPULATIONS 573 



Appendix IV 



TYPICAL EXACT CALCULATION 



A. Original Specification 



The exact expression (A5) for the probabiUty of a correct selection 

 for any configuration simphfies if the configuration is least favorable. 

 For any pair of integers (j,n) we define 



b,, = P{Xa) = j\ = CyVpfiO'Cgm)""' (0 ^ J ^ n) (A46) 



b^j = P{X,,y = j] = CrivU - d*yigii} + d*r^' {O^j^ n) (A47) 



B,j = P{X(2) ^ j} (A48) 



Then the exact probability Pes of a correct selection for the least 

 favorable configuration can be written as 



Pes = j: hj Z -%^. bh B^ (A49) 



3=0 1=0 L -\- I 



where B2,-i is defined to be zero. Here, for each value of X(i) , the letter 

 i denotes the number of processes that tie with X(i) for first place and 

 for any given value of i the conditional probability of a correct selection 

 is 1/(1 + i). Taking k = 4 as a typical case, we can write (A49) more 

 explicitly as 



n Q n n 



Pes = Z^ bijB^j-i + ^ Z^ bijbijBij-i + 2^ bijb2jB2j-i 



(A^O) 

 1 " 



+ 1 S K^lj 



4 j=o 



If n ^ 10 then we may use the symmetric configuration, i.e., we may set 

 Pui = 2 (1 + d*), in computing from (A49) or (A50). 



B. Alternative Specification 



The probability P^s of a correct selection for the alternative specifi- 

 cation is the same as in (A49) and (A50) except that we now define 



bii = P{X(,) = j} = C/'(pti,y(qti,r~\i = 1, 2) (A51) 



B2J = P{Zc2) ^ j] (A52) 



A typical exact calculation for A; = 4, using (A50), (A51) and (A52) with 



Pm = P[i] = 0.75 



