10 candidates for a job are interviewed
Source: Chinese Mathematical Olympiad 2003 Problem 5
February 18, 2012
probabilitycombinatorics proposedcombinatorics
Problem Statement
Ten people apply for a job. The manager decides to interview the candidates one by one according to the following conditions:
i) the first three candidates will not be employed;
ii) from the fourth candidates onwards, if a candidate's comptence surpasses the competence of all those who preceded him, then that candidate is employed;
iii) if the first nine candidates are not employed, then the tenth candidate will be employed.
We assume that none of the applicants have the same competence, and these competences can be ranked from the first to tenth. Let represent the probability that the th-ranked applicant in competence is employed. Prove that:
i) ;
ii)
iii) .Su Chun