57 lines
1.8 KiB
TeX
57 lines
1.8 KiB
TeX
\section{The Secretary Problem}
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\definition{}
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Say we need to hire a secretary. We have exactly one position to fill,
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and we must fill it with one of $n$ applicants. These $n$ applicants,
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if put together, can be ranked unambiguously from \say{best} to \say{worst}.
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\vspace{2mm}
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We interview applicants in a random order, one at a time. \par
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At the end of each interview, we either reject the applicant (and move on to the next one),
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or select the applicant (which fills the position and ends the process).
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\vspace{2mm}
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Each applicant is interviewed exactly once---we cannot return to an applicant we've rejected. \par
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In addition, we cannot reject the final applicant, as doing so will leave us without a secretary.
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\vspace{2mm}
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For a given $n$, we would like to maximize our probability of selecting the best applicant. \par
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This is called the \textit{secretary problem}.
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\problem{}
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If $n = 1$, what is the best hiring strategy, and what is the probability that we hire the best applicant?
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\vfill
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\problem{}
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If $n = 2$, what is the best hiring strategy, and what is the probability that we hire the best applicant? \par
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Is this different than the probability of hiring the best applicant at random?
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\vfill
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\pagebreak
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\problem{}
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What happens if $n = 3$?
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\begin{itemize}
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\item Find the probability of hiring the best applicant at random
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\item Find the best possible hiring strategy. \par
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What is its probability of success?
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\end{itemize}
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\hint{In this case, three is a fairly small number. \par
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It is easy to show that a strategy is optimal by considering all cases.}
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\vfill
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\problem{}
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Should we ever consider hiring a candidate that \textit{isn't} the best we've seen so far? \par
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Why or why not? \hint{Read the problem again.da}
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\vfill
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\remark{}
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To find the optimal solution to the secretary problem, we'll restrict ourselves to |