[Eng.]John Haigh: “The Mathematics of Chance,” translated by Li Daqiang, Jilin People’s Publishing House, August 2001, 26 yuan
Pages191~192 The Secretary Selection Problem
Suppose you want to hire a secretary. A group of applicants is waiting in line outside your office for interviews. You interview them one by one. At the end of each interview, you have two choices:1.reject this applicant and interview the next person;2.decide to hire the current applicant and end the recruitment process. You must make your decision on the spot. Once you have rejected someone, you cannot hire that person. How can you maximize the probability of hiring the best applicant?
In fact, some examples in life are similar to this. For instance, you arrive in an unfamiliar small town. You need to choose a restaurant from a row of little eateries for lunch; how should you choose in the most reasonable way? Stop at one restaurant to eat, or keep walking?
Your goal is to maximize the probability of hiring the best applicant. Suppose there are60 applicants; if you choose at random, you have a 1/60 chance of picking the right person. Of course, you can do better than that. The usual strategy is this: reject the first portion of the applicants, then examine the later ones; if an applicant appears whose qualifications are better than all the previous applicants’, hire that person and end the recruitment. The worst-case scenario is that you find that the later applicants are always worse than some applicant in the initial group, so that you end up having to hire the last applicant—who may be quite poor. Now the crucial question is how to determine the proportion of people you must definitely reject. First consider the most obvious strategy: reject the first half of the applicants, and thereafter, if an applicant appears whose qualifications are better than those of all previous applicants, hire that person. With this strategy, the probability of finding the best candidate is a little more than1/4.
Let us analyze this strategy. Consider the positions of two applicants in the whole line: the best applicant and the second-best applicant. If the second-best applicant is in the first half of the line and the best applicant is in the second half, then you are sure to get the best candidate. This conclusion is very obvious. In other cases, you will not get the best candidate. The probability that the second-best applicant is in the first half of the line is 1/2; on that premise, the probability that the best applicant is in the second half is a little more than 1/2, because one position in the first half has already been taken by the second-best applicant. So the probability that you get the best candidate is a little more than 25%.
In fact, one-half is not the optimal proportion. We can raise the probability of getting the best candidate to 37 %. The method is: reject the first 37 % of the applicants, and among the later applicants choose one who is better than all the earlier ones. Below are the details of the solution; readers who are interested may carefully ponder this passage. When you are choosing a restaurant in an unfamiliar small town, you might as well use the same strategy.
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I’ll omit the solution. The exact answer should be 1/e (e, of course, being the legendary base of the natural logarithm), which is approximately 37%.
This problem is dedicated to those classmates looking for jobs—for example, if you decide to apply to at most 100 employers, or if you decide to find a job within one year, then you can treat the first 37 employers, or the first four and a half months, as the trial run…
Of course, the actual situation is certainly different from a math problem. If you see a company that satisfies you, you should seize the opportunity, because if the best choice appears within the first 37%, then following this method you will inevitably have to wait until the very last moment to make a decision. And, even more importantly, you still have to consider whether the employer wants you…
Also, if you think the boss in charge of hiring has done this problem too, and is also hiring according to this method, then it is best to aim for a position around 38%–40%~~~
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Translated from the Chinese original with AI assistance. The original text is authoritative.
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