% use [nosolutions] flag to hide solutions.
% use [solutions] flag to show solutions.
\documentclass[
	solutions,
	singlenumbering,
]{../../resources/ormc_handout}

\usepackage{multicol}
\usepackage{tikz}
\usepackage{graphicx}
\graphicspath{ {.} }

\newcommand{\qe}{\stackrel{?}{=}}
\newcommand{\qgt}{\stackrel{?}{>}}
\newcommand{\qlt}{\stackrel{?}{<}}

\uptitlel{Advanced 2}
\uptitler{Fall 2022}
\title{Euler's Number}
\subtitle{
	Prepared by Mark on \today. \\
	Based on a handout by Oleg Gleizer and Olga Radko
}


\begin{document}

	\maketitle

	The goal of this mini-course is to construct Euler's number, one of the most important constants in mathematics, physics, economics, and finance. Make sure you fully understand all definitions before trying to solve problems that use them.

	\section{Compound Interest}

	Let $P$ be an amount of primary capital which is invested at an annual rate $r$. \par
	Let $V(t)$ be the value of the investment in $t$ years.

	\problem{}
	Derive the formula for $V(t)$ if the annual rate $r$ is compounded yearly. \par
	Derive the formula for $V(t)$ if the annual rate $r$ is compounded monthly. \par
	Derive the formula for $V(t)$ if the annual rate $r$ is compounded $n$ times a year, $n \in \mathbb{N}$. \par
	\hint{\say{Compound monthly} means that $\frac{1}{12}$ of $r$ is applied every month.}

	\begin{solution}
		$V(t) = P(1 + \frac{r}{n})^{nt}$
	\end{solution}

	Next, let's try to understand how we can compound interest continuously.

	\vfill
	\pagebreak



	\input{parts/1 limits.tex}
	\input{parts/2 e.tex}
	\input{parts/3 more e.tex}



	\section{$e$ and Probability}

	\problem{}
	A gambler plays a game $n$ times. Each time he plays, his chance of winning is $p$. What are the odds he will win exactly $k$ times?

	\vfill

	\problem{}
	A gambler plays a game $10,000$ times. Each time he plays, he has a $\frac{1}{10,000}$ chance of winning. What are the odds he loses every time?

	\vfill

	\problem{}
	$n$ people participate in a gift exchange. Each person puts their name in a hat, then names are drawn at random. For a large $n$, what is the probability that someone will draw their own name?

	\vfill
	\pagebreak

\end{document}
