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+\documentclass[a4paper]{article} +\usepackage{array} +\usepackage{adjustbox} +\usepackage{graphicx} +\usepackage[left=3cm,right=3cm,top=3cm, bottom=3cm]{geometry} +\usepackage{amsmath} +\usepackage{amssymb} +\usepackage{caption} +\usepackage[headsepline, footsepline]{scrlayer-scrpage} +\usepackage{enumerate} +\usepackage{dsfont} +\usepackage{float} +\usepackage{ifthen} +\usepackage[]{mathtools} +\usepackage[]{mathbbol} +\usepackage{multicol} +\usepackage{enumitem} +\usepackage[hidelinks]{hyperref} +\usepackage[]{circuitikz} +\usepackage{tcolorbox} % Für schöne Boxen +\usetikzlibrary{circuits.logic.IEC} +\usepackage{tabularray} +\usepackage{tikz} +\usepackage{tkz-graph} +\usepackage{listings} +\usepackage{xcolor} +\usepackage{setspace} +\usepackage{letltxmacro} +\usepackage{wrapfig} + +\usetikzlibrary{calc} + +\setlist{itemsep=0.3em, topsep=0.5em, parsep=0pt} + +\newcommand{\bpf}[1]{% + \par\vspace{0.8\baselineskip}% Abstand vor der Überschrift + \noindent\textbf{#1}% Fettgedruckte Überschrift + \par\vspace{0.3\baselineskip}% Abstand nach der Überschrift +} + + + +\renewcommand{\contentsname}{Inhaltsverzeichnis} +\renewcommand{\figurename}{Grafik} +\renewcommand{\partname}{Teil} +\renewcommand{\epsilon}{\varepsilon} + +\definecolor{darkgrey}{HTML}{232327} + +% Zählerdefinition mit AUTOMATISCHEM SUBSECTION-RESET +\newcounter{commoncounter}[subsection] +\renewcommand{\thecommoncounter}{\thesection.\arabic{commoncounter}} + +\usepackage{xparse} + +\newcommand{\answer}[1]{% + \ifhmode\newline\newline\fi + \foreach \n in {1,...,#1}{\noindent\rule{\linewidth}{0.4pt}\par\vspace{1em}} +} + + +\newcommand{\currentsection}{} +\newcommand{\currentsubsection}{} + +\LetLtxMacro{\oldsection}{\section} +\LetLtxMacro{\oldsubsection}{\subsection} + +\RenewDocumentCommand{\section}{s o m}{% + \IfBooleanTF{#1}% + { \oldsection*{#3} }% Stern-Variante → unverändert + {% + \IfValueTF{#2}% + { \oldsection[#2]{#3} }% Optionales TOC-Argument + { \oldsection{#3} }% + \renewcommand{\currentsection}{#3}% + }% +} + +\RenewDocumentCommand{\subsection}{s o m}{% + \IfBooleanTF{#1}% + { \oldsubsection*{#3} }% + {% + \IfValueTF{#2}% + { \oldsubsection[#2]{#3} }% + { \oldsubsection{#3} }% + \renewcommand{\currentsubsection}{#3}% + }% +} + + + +% Zähler direkt bei Dokumentstart initialisieren +\AtBeginDocument{\setcounter{commoncounter}{0}} + +% BOX-DEFINITIONEN mit korrigierter Zählerlogik +\newtcolorbox{definitionbox}[1][]{ + before title={\refstepcounter{commoncounter}}, % KRITISCH: vor dem Titel! + title={Definition \thecommoncounter: #1}, + colback=white, + colframe=white!75!darkgrey, + fonttitle=\bfseries, + boxrule=0.6mm, + coltitle=black, + rounded corners, + before skip=10pt, + after skip=10pt +} + +\newtcolorbox{extrataskbox}[1][]{ + before title={\refstepcounter{commoncounter}}, % KRITISCH: vor dem Titel! + title={Zusatzaufgabe: #1}, + colback=white, + colframe=white!75!red, + fonttitle=\bfseries, + boxrule=0.6mm, + coltitle=black, + rounded corners, + before skip=10pt, + after skip=10pt +} + +\newtcolorbox{merkbox}[1][]{ + title={Merken: #1}, + colback=white, + colframe=white!75!red, + fonttitle=\bfseries, + boxrule=0.6mm, + coltitle=black, + rounded corners, + before skip=10pt, + after skip=10pt +} + + + +\definecolor{codegreen}{rgb}{0,0.6,0} +\definecolor{codeblue}{rgb}{0,0,0.8} +\definecolor{codered}{rgb}{0.8,0,0} +\definecolor{lightgray}{rgb}{0.95,0.95,0.95} + +\lstdefinestyle{CStyle}{ + language=C, + basicstyle=\ttfamily\small, % Monospace-Schrift + keywordstyle=\color{blue}\bfseries, % Schlüsselwörter fett und blau + stringstyle=\color{red}, % Strings rot + commentstyle=\color{codegreen}, % Kommentare grün + backgroundcolor=\color{lightgray}, % Hintergrundfarbe + numbers=left, % Zeilennummern links + numbersep=10px, % Abstand zwischen Zeilennummern und Code + numberstyle=\color{gray}\texttt, + stepnumber=1, % Zeilennummerierung Schrittweite 1 + frame=single, % Rahmen um den Code + tabsize=4, % Tabulatorgröße + breaklines=true, % Zeilenumbruch aktivieren + captionpos=below, + showstringspaces=false, + xleftmargin=15pt, % Linker Rand für den Code (verschiebt alles nach rechts) +} + + +\lstdefinestyle{ShellStyle}{ + language=Bash, + basicstyle=\ttfamily\small, % Monospace-Schrift + keywordstyle=\color{black}, % Schlüsselwörter fett und blau + stringstyle=\color{black}, % Strings rot + commentstyle=\color{codegreen}, % Kommentare grün + backgroundcolor=\color{lightgray}, % Hintergrundfarbe + numbers=left, % Zeilennummern links + numbersep=10px, % Abstand zwischen Zeilennummern und Code + numberstyle=\color{gray}\texttt, + stepnumber=1, % Zeilennummerierung Schrittweite 1 + frame=single, % Rahmen um den Code + tabsize=4, % Tabulatorgröße + breaklines=true, % Zeilenumbruch aktivieren + captionpos=below, + showstringspaces=false, + xleftmargin=15pt, % Linker Rand für den Code (verschiebt alles nach rechts) +} + + +% Umgebung für Listings mit Titel und Zähler +\newenvironment{codeexample}[1][]{ + \refstepcounter{commoncounter} % Zähler erhöhen + \lstset{ + style=CSharpStyle, + caption={Listing \thecommoncounter: #1}, % Titel mit Zähler + label={listing:\thecommoncounter} + } +}{} + +\makeatletter +% Internen Zähler von subsubsection auf commoncounter umbiegen +\let\c@subsubsection\c@commoncounter + +% Anzeige der Nummer angleichen +\renewcommand{\thesubsubsection}{\thecommoncounter} +\makeatother + +\setlength{\parindent}{0pt} + + + +\newcommand{\curdate}{09.07.2026} +\newcommand{\curtitle}{Galton x Plinko} + +\title{\curtitle} +\author{Fierke, Janik, Weidlich} +\pagestyle{scrheadings} +\date{\curdate} +\begin{document} + \ihead{\normalfont\normalsize Mathematik} + \chead{\normalfont\normalsize \curtitle} + \ohead{\normalfont\normalsize \curdate} + \ofoot{\normalfont\normalsize Seite {\pagemark} von \pageref{LastPage}} + \ifoot{\normalfont\normalsize Fierke, Janik, Weidlich} + + \section[2]{\curtitle} + + { + \begin{wrapfigure}{r}{0.2\textwidth} + \centering + \includegraphics[width=0.7\linewidth]{eurox2.png} + + \end{wrapfigure} + + Plinko ist ein Glücksspiel aus den 1980er Jahren. Hierbei lässt man einen Ball von oben auf Stäbchen fallen, die wie eine Pyramide angeordnet sind. Ganz unten gibt es Fächer, bei denen der eingesetzte Geldbetrag abhängig vom Fach mit einem bestimmten Faktor multipliziert wird. + + \begin{enumerate} + + \item Probiert dich auf der Webseite {\color{blue}\hyperref{https://plinko-web-game.netlify.app/}{}{}{plinko-web-game.netlify.app}} einmal aus! + + + \end{enumerate} + } + + \begin{enumerate} + \setcounter{enumi}{1} + \item Stell eine Vermutung auf, wieso in der Mitte aller Fächer die kleinsten Faktoren liegen. + \answer{3} + + \item Stell eine Vermutung auf, wieso die Faktoren an den Randfächern am größten sind. + \answer{3} + + \item Argumentiere, wieso man auf lange Sicht verliert oder gewinnt. + \answer{3} + \end{enumerate} + + Mit dem Wissen über das Galtonbrett und seine Verteilung können wir nun fundierte Schlüsse über Plinko ziehen: + \begin{enumerate} + \item Bestimme die Wahrscheinlichkeit mit der eine Kugel in Plinko nach links bzw. rechts hüpft. + \answer{1} + \item Nenne die Anzahl der Ebenen in Plinko. \hrulefill\\ + \item Gib die Anzahl an Faktoren (Fächern) an. \hrulefill\\ + \item Gib nun eine Zufallsvariable $X\sim \text{Bin}_{n,p}$ an. \hrulefill\\ + \item Berechne die Wahrscheinlichkeit folgender Ereignisse: + \begin{enumerate}[label=\alph*)] + \item Die Kugel landet beim Multiplikator $0,3x$. + \item Eine Kugel holt ihren gesetzten Betrag exakt wieder rein. + \item Eine Kugel landet in einem der beiden höchsten Faktoren. + \item Zwei Kugeln holen mit einem Einsatz von je 1\$ einen Gewinn von insgesamt 4\$ + \end{enumerate} + \item Bestimme den Erwartungswert, wenn der Einsatz immer 1\$ ist. + \item Überprüfe die These: "\textit{Je höher mein Einsatz ist, desto mehr bekomme ich raus.}". + + \end{enumerate} + + + \label{LastPage} +\end{document}