From c09fc059ea38a367c43c8647a6e86c65cb41984c Mon Sep 17 00:00:00 2001 From: Elias Fierke Date: Thu, 16 Jul 2026 09:36:46 +0200 Subject: [PATCH] =?UTF-8?q?L=C3=B6sungsblatt=20f=C3=BCr=20ab2?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit --- material/ab2_Lös.pdf | Bin 0 -> 111956 bytes material/ab2_Lös.tex | 293 +++++++++++++++++++++++++++++++++++++++++++ 2 files changed, 293 insertions(+) create mode 100644 material/ab2_Lös.pdf create mode 100644 material/ab2_Lös.tex diff --git a/material/ab2_Lös.pdf b/material/ab2_Lös.pdf new file mode 100644 index 0000000000000000000000000000000000000000..1a6b3e64c7c933cf0f255f4b77b6101804391009 GIT binary patch literal 111956 zcmcG!Q;aA;v?bcM-KTBaIBnauZQHhO+qP}nw%v2)-ghTAnMvN{C9ggzwRfs^;b*V4 zSCPmGiO?|6vO$r|Ee)?hF|gv(N0LQ4PEg7^DgOPqHUN|;A{chaCZ$6PiyijaZ;gbcm9ZW%>Z zynCxjDeVuN015{iY>M+&TNYc1Nhxym;#=ng^Jz*(dD$h>m;-KVb*_u@%J&x^{1Z4* 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+\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{satzbox}[1][]{ + before title={\refstepcounter{commoncounter}}, % KRITISCH: vor dem Titel! + title={Satz \thecommoncounter: #1}, + colback=white, + colframe=white!75!yellow, + 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} + + \setcounter{section}{1} + \section{\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.\\ + \textit{\color{olive}Weil in der Mitte die meisten Kugeln hinfallen werden.}\\ + + \item Stell eine Vermutung auf, wieso die Faktoren an den Randfächern am größten sind.\\ + \textit{\color{olive}Weil dort die wenigsten Kugeln hinfallen werden.}\\ + + \item Argumentiere, wieso man auf lange Sicht verliert oder gewinnt.\\ + \textit{\color{olive}individuell} + \end{enumerate} + + Mit dem Wissen über das Galtonbrett und seine Verteilung können wir nun fundierte Schlüsse über Plinko ziehen: + \begin{enumerate}[itemsep=11pt] + \setcounter{enumi}{4} + \item Die Wahrscheinlichkeit mit der eine Kugel in Plinko nach links bzw. rechts hüpft ist \textit{\color{olive}50\%} + \item Nenne die Anzahl der Ebenen in Plinko. \textit{\color{olive}16} + \item Gib die Anzahl an Faktoren (Fächern) an. \textit{\color{olive}17} + \item Gib nun eine Zufallsvariable $X\sim \text{Bin}_{n,p}$ an.\\ \textit{\color{olive}$X\sim\text{Bin}_{16,0,5}$ mit $\mathbb{P}(X=k)=\binom{16}{k}\cdot 0.5^{16}$, wobei $k=\{0,\dots,16\}$} + \item Berechne die Wahrscheinlichkeit folgender Ereignisse: + \begin{enumerate}[label=\alph*),itemsep=11pt] + \item Die Kugel landet beim Multiplikator $0,3x$. \textit{\color{olive}Faktor $0,3x$ entspricht dem Fach 8.\\ + $\mathbb{P}(X=8)=\binom{16}{8}\cdot 0.5^{16}=0.196380615234375\approx19,6\%$} + \item Eine Kugel holt ihren gesetzten Betrag exakt wieder rein. \textit{\color{olive}Die Kugel trifft ein Fach mit dem Faktor $1x$, also Fach 6 oder 10.\\ + $\mathbb{P}(X=6)+\mathbb{P}(X=10)=\left(\binom{16}{6}+\binom{16}{10}\right)\cdot 0.5^{16}=0.244384765625\approx24,4\%$} + \item Eine Kugel landet in einem der beiden höchsten Faktoren. \textit{\color{olive}Die Kugel trifft Fach 0 oder 16.\\$\mathbb{P}(X=0)+\mathbb{P}(X=16)=\left(\binom{16}{0}+\binom{16}{16}\right)\cdot 0.5^{16}=0.000030517578125\approx0,003\%$} + % \item Zwei Kugeln holen mit Einsatz von je 1\$ einen Gewinn von gesamt 4\$. \hrulefill + \end{enumerate} + \item Bestimme den Erwartungswert, wenn der Einsatz immer 1\$ ist. + \textit{\color{olive}\begin{align*} + \mathbb{E}(X)&=110\$\cdot\mathbb{P}(X=0)+41\$\cdot\mathbb{P}(X=1)+\dots+41\$\cdot\mathbb{P}(X=15)+110\$\cdot\mathbb{P}(X=16)-1\$\\ + &=-0.0101165771484375 + \end{align*}} + \item Überprüfe die These: "\textit{Je höher mein Einsatz ist, desto mehr bekomme ich raus.}". + \textit{\color{olive}Zum Einen ist die These richtig, da man bei einem Hauptgewinn mehr gewinnt, als wenn + man weniger gesetzt hätte. Man kann jedoch auch deutlich mehr verlieren. Unter + Betrachtung des Erwartungswertes werden die Auszahlungssummen gemäß des + Einsatzes mutlipliziert. Das heißt $k\cdot\mathbb{E}(X)$ wenn $k$ der Einsatz ist. Da $\mathbb{E}(X)<0\Rightarrow k\cdot\mathbb{E}(X)<\mathbb{E}(X)\text{ wenn }k>1$} + + \end{enumerate} + + + \label{LastPage} +\end{document}