\section{The Secretary Problem}

\definition{}
Say we need to hire a secretary. We have exactly one position to fill,
and we must fill it with one of $n$ applicants. These $n$ applicants,
if put together, can be ranked unambiguously from \say{best} to \say{worst}.

\vspace{2mm}

We interview applicants in a random order, one at a time. \par
At the end of each interview, we either reject the applicant (and move on to the next one),
or select the applicant (which fills the position and ends the process).

\vspace{2mm}

Each applicant is interviewed exactly once---we cannot return to an applicant we've rejected. \par
In addition, we cannot reject the final applicant, as doing so will leave us without a secretary.

\vspace{2mm}

For a given $n$, we would like to maximize our probability of selecting the best applicant. \par
This is called the \textit{secretary problem}.


\problem{}
If $n = 1$, what is the best hiring strategy, and what is the probability that we hire the best applicant?

\vfill

\problem{}
If $n = 2$, what is the best hiring strategy, and what is the probability that we hire the best applicant? \par
Is this different than the probability of hiring the best applicant at random?

\vfill
\pagebreak


\problem{}
What happens if $n = 3$?
\begin{itemize}
	\item Find the probability of hiring the best applicant at random
	\item Find the best possible hiring strategy. \par
	What is its probability of success?
\end{itemize}
\hint{In this case, three is a fairly small number. \par
It is easy to show that a strategy is optimal by considering all cases.}

\vfill

\problem{}
Should we ever consider hiring a candidate that \textit{isn't} the best we've seen so far? \par
Why or why not? \hint{Read the problem again.da}

\vfill

\remark{}
To find the optimal solution to the secretary problem, we'll restrict ourselves to