Suppose a busy manager must hire an assistant. The manager is so busy that there will be only one opportunity to interview each candidate. At the end of each interview the manager must decide to hire the current candidate or to dismiss them and interview the next applicant for the assistant position.
Before any of the interviews start, the manager has no idea the range of abilities represented in the applicant pool. The applicants will be interviewed in a random order.
How can the manager maximize the probability of hiring the most qualified applicant for the assistant's position?
&circlef; There are N applicants for the assistant's position of varying abilities.
Since the interviewer does not know anything about the spectrum of abilities represented in the candidate pool, one strategy would be to sample the applicant pool by interviewing k candidates (0 < k < N) and then hiring the next applicant to exceed the maximum aptitude found among the first k candidates.
This strategy of course fails if the highest qualified applicant is among the k candidates sampled. This strategy does not also guarantee that the best candidate will be hired. It may happen that an applicant who is not the highest qualified, but whose abilities exceed those of the first k candidates, will be interviewed before the most qualified person, and thus will be hired and the most qualified applicant will never be interviewed.
If there are N candidates, the probability that any particular applicant is the most qualified is . The probability that the best candidate is not among the k sampled is 1 - .
If the most qualified applicant should be the k+ (the probability of this event is ), then the probability that the best candidate will be hired is .
Suppose the most qualified applicant is the k+ The probability that the k+ candidate is better than the first k candidates is . The probability of hiring the best candidate in this case is .
To continue this line of reasoning, suppose the most qualified applicant is the k+ The probability the k+ candidate is not hired is . If the k+ applicant is more qualified than any of the first k people interviewed, then candidate number k+2 is also more qualified than applicant k+1, else the k+ candidate would have been hired. The probability that the k+ applicant is not better than the first k+1 candidates is . Thus the probability of hiring the most qualified applicant is then = .
Thus in general if the best candidate is the k+ applicant (0 < i ≤N-k), the probability they will be hired is .
The probability of hiring the best candidate from among the N when the manager samples k candidates is
P(k) = + + + ⋯ + = .
How do we maximize this probability?
Suppose there are 30 candidates that could be interviewed. The probability of hiring the most qualified candidate after interviewing k applicants is shown in the next graph.
The maximum probability occurs when 11 candidates are sampled and then the next applicant to be interviewed who exceeds the qualifications of the first 11 is hired.
Now suppose there are between 2 and 50 candidates to be interviewed.
We need to know for each value of N what the optimal value of k is.
According to this graph, the manager should sample approximately 36% of the applicants. The corresponding probabilities are given in the next graph.