You receive 15 applications for a place in a mathematics competition and plan to interview all 15 candidates in a random order. Since your time is limited you have decided to use the following strategy for the interview: Reject the first $$k-1$$ candidates and then accept the first of the following candidates who is better than all those previously interviewed. If no such candidate appears then select the last candidate. Once a candidate is accepted no further interviews take place.
What is the optimal number $$k$$ for which you have the highest probability of selecting the best candidate?