ISL 1988 Problem 81

There are $n \geq 3$ job openings at a factory, ranked $1$ to $n$ in order of increasing pay. There are $n$ job applicants, ranked from $1$ to $n$ in order of increasing ability. Applicant $i$ is qualified for job $j$ if and only if $i \geq j.$ The applicants arrive one at a time in random order. Each in turn is hired to the highest-ranking job for which he or she is qualified AND which is lower in rank than any job already filled. (Under these rules, job $1$ is always filled, and hiring terminates thereafter.) Show that applicants $n$ and $n - 1$ have the same probability of being hired.