# A simple probability problem

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.

My reasoning:

• There are $n*(n-1)*(n-2)$ many ways the interviewers put a different person at the first position.

• So, $n^3 - n*(n-1)*(n-2)$ ways that at least two interviewers will put the same person at the first position

• So, $\frac{n^3 - n*(n-1)*(n-2)}{n^3} = \frac{3n^2-2n}{n^3}$ is the desired probability

But the answer given is $\frac{3n-2}{n^3}$

Where is the problem in my reasoning? thanks.

The probability that the second interviewer puts a different person at the first position is

$$\frac{n-1}{n}\,.$$

The probability that the third interviewer puts a different person at the first position is

$$\frac{n-2}{n}\,.$$

Hence, the probability that there are three different persons at the first position is

$$p(\text{all different})=\frac{n-1}{n}\cdot \frac{n-2}{n} = \frac{n^2-3n+2}{n^2} \,.$$

The contrary of three different persons at the first position is not three different persons at the first position (i.e., at least two times the same person is at position one). Therefore, we calculate

$$p(\text{at least two identical}) = p(\lnot\text{all different}) = 1 - p(\text{all different}) =\\1-\frac{n^2-3n+2}{n^2} = \frac{3n-2}{n^2}\,.$$

• Hi Sven, you have found exactly the same answer with me, but the given answer in the text book is different, note the $n^3$ at the denominator. – Sanyo Mn Sep 6 '17 at 11:53
• Might still be a typo. – bayerj Sep 6 '17 at 11:54
• You can understand the question as relating to a specific candidate being selected and not just any candidate. In this case you have to multiply your result by 1/n. – Zahava Kor Sep 6 '17 at 16:34