Math 2.0

Probability theory and stochastic processes Stochastic processes: general theory Problem

Secretary problem

Builds on Stopping timePermutation

Problem

Let n≥1n \ge 1 applicants with distinct qualities be interviewed one at a time, in an order given by a uniformly random permutation. After each interview, the only information is the relative rank of the current applicant among those seen so far. After each interview the applicant must be accepted or rejected at once, and the decision cannot be changed. The process ends at the first acceptance. Find a rule that maximises the probability of accepting the best of the nn applicants, and find this maximal probability. A rule is a stopping time for the observed relative ranks.

Remarks

The problem is solved, and it is a standard example of optimal stopping. An optimal rule has a threshold form: reject the first r−1r - 1 applicants, then accept the first applicant who is better than all earlier ones. For r≥2r \ge 2 this rule picks the best applicant with probability Pn(r)=r−1n∑i=rn1i−1,P_n(r) = \frac{r - 1}{n} \sum_{i=r}^{n} \frac{1}{i - 1}, and Pn(1)=1/nP_n(1) = 1/n. The optimal rr satisfies r/n→1/er/n \to 1/e and the maximal probability tends to 1/e≈0.3681/e \approx 0.368 as n→∞n \to \infty. For n=3n = 3, the rule with r=2r = 2 succeeds with probability 13(1+12)=12\frac{1}{3}(1 + \frac{1}{2}) = \frac{1}{2}, which is optimal.

Reading path

To understand this page from the ground up, read these 3 pages in order. The foundations come first.

Show the path as a graph
Permutation, concept, no evidence levelPermutationconceptStopping time, concept, no evidence levelStopping timeconceptSecretary problem, problem, no evidence levelSecretary problemproblem
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Permutation concept
  2. Stopping time concept
  3. Secretary problem problem

Further reading

  1. Web https://en.wikipedia.org/wiki/Secretary_problem
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Kindproblem
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MSC60G40, 62L15
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UsesStopping time, Permutation
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Changed2 November 2026

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