Suppose that I have to interview $N$ many candidates for a job. Each candidate is assigned a "competency" score $C_i$ after being interviewed (where $C_i$ is the score of the $i$th candidate).
The rules of the interviewing process are as follows:
- Candidates are interviewed in random order until one is chosen.
- If candidate $i$ is chosen/hired, he or she must be better (or equal) than all the previously interviewed candidates. That is, $$C_i \geq C_{i-k}$$ for all meaningful $k$.
- You cannot go back to a previous candidate (if you move on, you cannot hire them later) and no one is interviewed twice.
- You only can hire one person.
What is the probability that if you hire the $m$th candidate that he or she is the best candidate? (i.e. has competency score $C_m \geq C_i$ for every index $i$).
This question was presented in one of my probability classes. I honestly am not sure if there is one true and correct answer. But how do I show that? Is there in fact a correct and identifiable answer to this question? There seem to be some issues with stopping times, etc.
If there is no correct answer to this question, reason why this is so. Obviously, if there is indeed a correct answer, also indicate as such.
Just so as to present some sort of answer, I suggested to my classmates that the probability of selecting the best candidate could not be worse than $\frac{1}{N-m+1}$. We know that if $m$ was hired, he or she was better (or no worse) than all the previously interviewed candidates. So person $m$ is really only in competition with $N-m$ other people for being the "best" candidate.