diff --git a/Advanced/Stopping Problems/main.tex b/Advanced/Stopping Problems/main.tex new file mode 100755 index 0000000..3184e8a --- /dev/null +++ b/Advanced/Stopping Problems/main.tex @@ -0,0 +1,24 @@ +% 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} \ No newline at end of file diff --git a/Advanced/Stopping Problems/parts/0 intro.tex b/Advanced/Stopping Problems/parts/0 intro.tex new file mode 100644 index 0000000..fb61b36 --- /dev/null +++ b/Advanced/Stopping Problems/parts/0 intro.tex @@ -0,0 +1,26 @@ +\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{} +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 \ No newline at end of file diff --git a/Advanced/Stopping Problems/parts/1 secretary.tex b/Advanced/Stopping Problems/parts/1 secretary.tex new file mode 100644 index 0000000..eef82bf --- /dev/null +++ b/Advanced/Stopping Problems/parts/1 secretary.tex @@ -0,0 +1,57 @@ +\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 \ No newline at end of file