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There are n applicants for the director of computing. The applicants are interviewed independently by each member of the three-person search committee and ranked from 1 to n. A candidate will be hired if he or she is ranked first by at least two of the three interviewers. Find the probability that a candidate will be accepted if the members of the committee really have no ability at all to judge the candidates and just rank the candidates randomly. In particular, compare this probability for the case of three candidates and the case of ten candidates.

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Answer:

for  n =  3

       [tex]Z(3) = 0.26[/tex]  

for  n =  10

       [tex]Z(3) = 0.028[/tex]

Step-by-step explanation:

from the question we are that

  The  number of applicant is n

  The  number of  people on the committee is [tex]N = 3[/tex]

 

Generally given that the applicant ranking is random the probability of a candidate being accepted is mathematically represented as

       [tex]Z(n) =  N  *  \frac{1}{n} *  \frac{1}{n}  *  \frac{n-1}{n}  + \frac{1}{n} *  \frac{1}{n} * \frac{1}{n}[/tex]

Now  for  n =  3

       [tex]Z(3) =  3  *  \frac{1}{3} *  \frac{1}{3}  *  \frac{3-1}{3}  + \frac{1}{3} *  \frac{1}{3} * \frac{1}{3}[/tex]      

         [tex]Z(3) = 0.26[/tex]            

Now  for  n =  10

       [tex]Z(3) =  3  *  \frac{1}{10} *  \frac{1}{10}  *  \frac{10-1}{10}  + \frac{1}{10} *  \frac{1}{10} * \frac{1}{10}[/tex]      

         [tex]Z(3) = 0.028[/tex]

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