Finish exercise 1
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@ -26,8 +26,7 @@
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\input{lib/latex-common/common.tex}
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\pgfplotsset{colorscheme/rocket}
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%TODO: Fix path
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\newcommand{\res}{src/template/res}
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\newcommand{\res}{src/2025-11-07/res}
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% \tikzstyle{every node}=[font=\small]
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% \captionsetup[sub]{font=small}
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@ -84,7 +83,7 @@
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\item Wiederholung der für die Aufgaben wichtigsten Teile
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der Theorie
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\end{itemize}
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\item Angesetzte Struktur
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\item Struktur der Tutorien
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\begin{table}
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\begin{tabular}{l||c}
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Abschnitt & Dauer \\\hline\hline
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@ -108,68 +107,138 @@
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\subsection{Theorie Wiederholung}
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% TODO: Replace slide content with relevant stuff
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\begin{frame}
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\frametitle{Theorie Wiederholung I}
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\begin{frame}{Ereignisse \& Laplace}
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\vspace*{-15mm}
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\begin{itemize}
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\item Ereignisse
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\begin{columns}
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\column{\kitthreecolumns}
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\begin{greenblock}{Zufallsvariablen (ZV)}%
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\begin{align*}
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\text{Ergebnisraum: } & \hspace{5mm} \Omega =
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\mleft\{ \omega_1, \ldots, \omega_N \mright\}\\
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\text{Ergebnis: } & \hspace{5mm} \omega_i\\
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\text{Ereignis: } & \hspace{5mm} A \subseteq \Omega
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\end{align*}
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\column{\kitthreecolumns}
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\begin{lightgrayhighlightbox}
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Beispiel: Würfeln mit einem Würfel
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\begin{align*}
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\Omega &= \mleft\{ 1, \ldots, 6 \mright\}\\
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A &= \mleft\{ 1, 6 \mright\}
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\end{align*}\\[1em]
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\vspace*{-12mm}
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\end{lightgrayhighlightbox}
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\begin{lightgrayhighlightbox}
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Beispiel: Würfeln mit zwei Würfeln
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\begin{align*}
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\Omega &= \mleft\{(i,j): i,j \in \mleft\{
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1,\ldots, 6 \mright\}\mright\} \\
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A &= \mleft\{ (1,1),(2,2), \ldots, (6,6) \mright\}
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\end{align*}
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\vspace*{-12mm}
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\end{lightgrayhighlightbox}
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\vspace*{0mm}
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\end{columns}\pause
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\item Laplace'sches Zufallsexperiment
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\begin{gather*}
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\text{Voraussetzungen: }\hspace{5mm} \left\{
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\begin{array}{l}
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\lvert\Omega\rvert \text{ endlich}\\
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P(\omega_i) = \frac{1}{\lvert\Omega\rvert}
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\end{array}
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\right.\\[1em]
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P(A) = \frac{\lvert A \rvert}{\lvert \Omega \rvert} =
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\frac{\text{Anzahl ``günstiger''
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Möglichkeiten}}{\text{Anzahl Möglichkeiten}}
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\end{gather*}
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\end{itemize}
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\end{frame}
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\begin{frame}{Kombinationen und Hypergeometrische\\ Verteilung}
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\begin{itemize}
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\item Kombinationen: Ziehen ohne zurücklegen, ohne
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Betrachtung der Reihenfolge
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\vspace*{5mm}
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\begin{columns}
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\column{\kitthreecolumns}
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\begin{gather*}
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\lvert C_N^{(K)} \rvert = \binom{N}{K} =
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\frac{N!}{(N-K)!K!}
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\end{gather*}
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\column{\kitthreecolumns}
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\begin{lightgrayhighlightbox}
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Beispiel: Wie viele mögliche Ergebnisse gibt
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es beim Lotto ``6 aus 49''?
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\vspace*{0mm}
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\begin{align*}
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\begin{array}{c}
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N = 49 \\
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K = 6
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\end{array} \hspace{5mm} \rightarrow
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\hspace{5mm} \binom{49}{6} = 13983816
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\end{align*}
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\vspace*{-8mm}
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\end{lightgrayhighlightbox}
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\end{columns}
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\pause
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\item Hypergeometrische Verteilung
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\begin{columns}
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\column{\kitthreecolumns}
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\begin{gather*}
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P_r = \frac{\binom{R}{r}\binom{N-R}{n-r}}{\binom{N}{n}}
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\end{gather*}
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\column{\kitthreecolumns}
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\begin{lightgrayhighlightbox}
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Beispiel: In einer Urne sind N Kugeln, davon
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R rot. Wie groß ist die Wahrscheinlichkeit
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beim ziehen von n Kugeln (ohne Zurücklegen)
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genau r rote zu erwischen?
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\end{lightgrayhighlightbox}
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\end{columns}
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\end{itemize}
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\end{frame}
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\begin{frame}{Zusammenfassung}
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\begin{columns}
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\column{\kitthreecolumns}
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\begin{greenblock}{Laplace'sches Zufallsexperiment}%
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\vspace*{-6mm}
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\begin{gather*}
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f_X(x) := \frac{d}{dx} F_X(x) \\
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P(X \le x) = F_X(x) = \int_{-\infty}^{x} f_X(t) dt \\
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E(X) = \int_{-\infty}^{\infty} x\cdot f_X(x) dx
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P(A) = \frac{\lvert A \rvert}{\lvert \Omega \rvert} =
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\frac{\text{Anzahl ``günstiger''
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Möglichkeiten}}{\text{Anzahl Möglichkeiten}}
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\end{gather*}
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\end{greenblock}
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\column{\kitthreecolumns}
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\begin{greenblock}{Important Equations}%
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\begin{greenblock}{Kombinationen}%
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\vspace*{-6mm}
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\begin{gather*}
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f_X(x) := \frac{d}{dx} F_X(x) \\
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P(X \le x) = F_X(x) = \int_{-\infty}^{x} f_X(t) dt \\
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E(X) = \int_{-\infty}^{\infty} x\cdot f_X(x) dx
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\lvert C_N^{(K)}\rvert = \binom{N}{K} =
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\frac{N!}{(N-K)!K!}
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\end{gather*}
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\end{greenblock}
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\end{columns}
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\begin{greenblock}{Normalverteilung}
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\begin{columns}
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\column{\kitonecolumn}
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\column{\kitthreecolumns}
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\begin{greenblock}{Hypergeometrische Verteilung}%
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\vspace*{-6mm}
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\begin{gather*}
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\text{Normalverteilung:} \hspace{8mm}
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f_X(x) = \frac{1}{\sqrt{2\pi\sigma^2}}
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e^{-\frac{(x - \mu)^2}{2\sigma^2}}
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P_R = \frac{\binom{R}{r}\binom{N-R}{n-r}}{\binom{N}{n}}
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\end{gather*}
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\column{\kitthreecolumns}
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\begin{figure}
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\centering
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\begin{tikzpicture}
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\begin{axis}[
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domain=-4:4,
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samples=100,
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width=11cm,
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height=6cm,
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ticks=none,
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xlabel={$x$},
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ylabel={$f_X(x)$}
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]
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\addplot+[mark=none, line width=1pt] {exp(-x^2)};
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\end{axis}
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\end{tikzpicture}
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\end{figure}
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\end{columns}
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\end{greenblock}
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\end{frame}
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\column{\kitonecolumn}
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\end{columns}
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\end{frame}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\subsection{Aufgabe}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\subsection{Aufgabe}
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% TODO: Replace slide content with relevant stuff
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\begin{frame}
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\frametitle{Aufgabe 1: Ergebnisraum \& Hypergeometrische\\ Verteilung}
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\begin{frame}
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\frametitle{Aufgabe 1: Ergebnisraum \&
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Hypergeometrische\\ Verteilung}
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Bei einem Kartenspiel erhält ein Spieler 5 Karten aus einem Deck
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von 52 Karten (bestehend aus
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@ -184,16 +253,47 @@
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\end{enumerate}
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% tex-fmt: on
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\end{frame}
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\end{frame}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\section{Aufgabe 2}
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\begin{frame}
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\frametitle{Aufgabe 1: Ergebnisraum \&
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Hypergeometrische\\ Verteilung}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\subsection{Theorie Wiederholung}
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Bei einem Kartenspiel erhält ein Spieler 5 Karten aus einem Deck
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von 52 Karten (bestehend aus
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13 Arten mit je 4 Farben). Wie groß ist die Wahrscheinlichkeit,
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dass der Spieler
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% TODO: Replace slide content with relevant stuff
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\begin{frame}
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% tex-fmt: off
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\begin{enumerate}[a{)}]
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\item mindestens ein Ass hat?\pause
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\begin{gather*}
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P(\text{mindestens ein Ass}) = 1 - P(\text{kein Ass})
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= 1 - \frac{\binom{4}{0}\binom{48}{5}}{\binom{52}{5}} \approx 0.341
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\end{gather*}\pause\vspace*{-5mm}
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\item genau ein Ass hat?\pause
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\begin{gather*}
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P(\text{genau ein Ass}) = \frac{\binom{4}{1}\binom{48}{4}}{\binom{52}{5}} \approx 0.299
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\end{gather*}\pause
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\item mindestens zwei Karten der gleichen Art (“Paar”) hat?\pause
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\begin{align*}
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P(\text{mindestens zwei gleiche Karten}) &= 1 - P(\text{alle Karten unterschiedlich}) \\
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&= 1 - \frac{\text{Anzahl Möglichkeiten mit nur unterschiedlichen Karten}}{\text{Anzahl Möglichkeiten}}\\
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&= 1 - \frac{\binom{13}{5}\cdot 4^5}{\binom{52}{5}} \approx 0.493
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\end{align*}
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\end{enumerate}
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% tex-fmt: on
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\end{frame}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\section{Aufgabe 2}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\subsection{Theorie Wiederholung}
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% TODO: Replace slide content with relevant stuff
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\begin{frame}
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\frametitle{Theorie Wiederholung II}
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\begin{gather*}
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@ -230,17 +330,18 @@
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\end{tikzpicture}
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\end{subfigure}
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\end{figure}
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\end{frame}
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\end{frame}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\subsection{Aufgabe}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\subsection{Aufgabe}
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% TODO: Replace slide content with relevant stuff
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\begin{frame}
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% TODO: Replace slide content with relevant stuff
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\begin{frame}
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\frametitle{Aufgabe 2: Variationen \& Permutationen}
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Aufgabe 2: Variationen \& Permutationen
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Ein Burgerrestaurant bietet verschiedene Burger mit den Zutaten Salat
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Ein Burgerrestaurant bietet verschiedene Burger mit den
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Zutaten Salat
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(S), Käse (K), Tomate (T)
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und Patty (P) an. Diese werden zufällig für die Zubereitung eines
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Burgers ausgewählt.
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@ -264,13 +365,13 @@
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\end{enumerate}
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% tex-fmt: on
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\end{frame}
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\end{frame}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\section{Zusammenfassung}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\section{Zusammenfassung}
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% TODO: Replace slide content with relevant stuff
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\begin{frame}
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% TODO: Replace slide content with relevant stuff
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\begin{frame}
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\frametitle{Zusammenfassung}
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\begin{gather*}
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@ -307,7 +408,7 @@
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\end{tikzpicture}
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\end{subfigure}
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\end{figure}
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\end{frame}
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\end{frame}
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\end{document}
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\end{document}
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