Q11. Pg97. 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.
A candidate can be picked first at least twice in two ways \[ \begin{aligned} P(\text{At least two members picked candidate}) &= P(\text{All three place candidate first}) + P(\text{Exactly two pick candidate}) \\ P(\text{All three place candidate first}) &= \frac{1}{n} * \frac{1}{n} * \frac{1}{n} \\ &= \frac{1}{n^3} \\ P(\text{Exactly two pick candidate}) &= \frac{1}{n} * \frac{1}{n} * \frac{n - 1}{n} * 3 \\ &= \frac{3*n - 3}{n^3} \end{aligned} \]
Therefore; \[ \begin{aligned} P(\text{At least two members picked candidate}) &= \frac{1}{n^3} + \frac{3*n - 3}{n^3} \\ &= \frac{3*n - 2}{n^3} \end{aligned} \]
3 candidates. That is n = 3 \[ \begin{aligned} P(n = 3) &= \frac{3*3 - 2}{3^3} \\ &= 0.259 \end{aligned} \]
10 candidates. That is n = 10 \[ \begin{aligned} P(n = 10) &= \frac{3*10 - 2}{10^3} \\ &= 0.0.28 \end{aligned} \] Therefore, as the number of candidates increase the chance of being picked for that position reduces.