Stopping problems draft

Advanced/Stopping Problems/main.tex

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+% use [nosolutions] flag to hide solutions.
+% use [solutions] flag to show solutions.
+\documentclass[
+	solutions,
+	singlenumbering,
+	unfinished
+]{../../resources/ormc_handout}
+\usepackage{../../resources/macros}
+
+\usepackage{units}
+
+\uptitlel{Advanced 2}
+\uptitler{\smallurl{}}
+\title{Stopping problems}
+\subtitle{Prepared by Mark on \today{}}
+
+\begin{document}
+
+	\maketitle
+
+	\input{parts/0 intro.tex}
+	\input{parts/1 secretary.tex}
+
+\end{document}

Advanced/Stopping Problems/parts/0 intro.tex

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+\section{Introduction}
+
+\generic{Setup:}
+Suppose we toss a 6-sided die $n$ times. \par
+It is easy to detect the first time we roll a 6. \par
+What should we do if we want to annouce the \textit{last}?
+
+\problem{}<lastl>
+Given $l \leq n$, what is the probability that the last $l$
+tosses of this die contain exactly one six?
+
+\vfill
+
+\problem{}
+For what value of $l$ is the probability in \ref{lastl} maximal?
+
+\vfill
+
+\problem{}
+Finish your solution: \par
+In $n$ rolls of a six-sided die, how do we (most reliably) detect the last time we roll a 6?
+
+\vfill
+
+
+\pagebreak

Advanced/Stopping Problems/parts/1 secretary.tex

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+\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