Unknown Number of Applicants
A major drawback for applications of the solution of the classical secretary problem is that the number of applicants must be known in advance. One way to overcome this problem is to suppose that the number of applicants is a random variable with a known distribution of (Presman and Sonin, 1972). For this model, the optimal solution is in general much harder, however. Moreover, the optimal success probability is now no longer around 1/e. Indeed, it is intuitive that there should be a price to pay for not knowing the number of applicants. However, in this model the price is high. Depending on the choice of the distribution of the optimal win probability is typically much lower than 1/e, and may even approach zero. Looking for ways to cope with this new problem led to the following approach and result:
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