\documentclass[
	solutions,
	shortwarning,
	singlenumbering,
	nopagenumber,
	nowarning
]{../../resources/ormc_handout}
\usepackage{../../resources/macros}
\geometry{top = 20mm}

\title{Warm-Up: Parking Problems}
\subtitle{Prepared by \githref{Mark} on \today.}

\begin{document}

	\maketitle



	\problem{A Minor Inconvenience}
	A group of eight friends goes out to dinner. Each drives his own car, checking it in with valet upon arrival.
	Unfortunately, the valet attendant forgot to tag the friends' keys. Thus, when the group leaves the restaurant,
	each friend is handed a random key.
	\begin{itemize}
		\item What is the probability that everyone gets the correct set of keys?
		\item What is the probability that each friend gets the wrong set?
	\end{itemize}

	\vfill


	\problem{Bimmer Parking}
	A parking lot has a row of 16 spaces, of which a random 12 are taken. \par
	Ivan drives a BMW, and thus needs two adjacent spaces to park. \par
	What is the probability he'll find a spot?

	\vfill
	\pagebreak


	\problem{Petty Bruins in Westwood}
	Suppose 6 students live on a one-way street which features six parallel parking spaces.
	Each student has their favorite spot, and will always check if that spot is free.
	If it is, they will park there. If it isn't, they will take the next free spot.
	Backing up is not an option.

	If each student chooses a \say{favorite} spot independently and at random...\par
	\begin{itemize}
		\item what is the probability that each student has a different favorite spot?
		\item what is the probability that all six people will park successfully?
	\end{itemize}



	\vfill



	\problem{Ship Packing}
	A certain harbor services ships that are exactly 10 meters long.
	These ships are equipped with transverse thrusters, and thus have
	no problem parking in parallel (a ship may fit in any gap that is
	at least 10 meters long).

	\vspace{2mm}

	There is an empty 100-meter span where these ships may be parked in parallel.
	Parking spaces are not marked---ships may pick a parking spot anywhere on this
	continuous 100-meter length.

	\vspace{2mm}

	As ships arrive, they pick a random available spot along this span. \par
	How many ships do we expect to fit on this 100-meter region?

	\hint{This problem is difficult, start with simpler cases.}



	\vfill
	\pagebreak


\end{document}