Finish Tutorium 1
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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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\frametitle{Kombinatorik}
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\vspace*{-18mm}
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\begin{itemize}
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\item Potenzmenge
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\vspace*{-2mm}
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\begin{columns}
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\column{\kitfourcolumns}
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\begin{align*}
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\mathcal{P}\mleft( \Omega \mright) = \mleft\{ A:
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A \subseteq \Omega \mright\} \hspace{10mm}
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\left(\text{``Menge aller
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Teilmengen von $\Omega$''}\right)
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\end{align*}
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\column{\kittwocolumns}
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\begin{lightgrayhighlightbox}
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Beispiel
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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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\end{gather*}
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\begin{figure}
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\centering
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\begin{subfigure}[c]{0.5\textwidth}
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\centering
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\Omega = \{ A, B, C \}
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\end{gather*}%
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\vspace*{-15mm}%
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\begin{align*}
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\mathcal{P}(\Omega) = \{ &\emptyset,
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\mleft\{ A \mright\}, \mleft\{ B \mright\},
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\mleft\{ C \mright\}, \mleft\{ A, B \mright\},\\
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&\mleft\{ A, C \mright\},
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\mleft\{ B, C \mright\}, \mleft\{ A, B, C \mright\} \}
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\end{align*}%
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\vspace*{-14mm}%
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\end{lightgrayhighlightbox}
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\end{columns}
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\vspace*{-3mm}
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\item \pause Variationen und Kombinationen
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\setlength\extrarowheight{2mm}
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\begin{table}
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\begin{tabular}{r||l|l}
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& Mit Zurücklegen & Ohne Zurücklegen
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\\\hline\hline Mit Reihenfolge
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(\textit{Variationen}) & $\lvert
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\widetilde{V}_N^{(K)} \rvert = N^K$ & $\lvert
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V_N^{(K)}\rvert = \frac{N!}{(N-K)!} $ \\\hline
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Ohne Reihenfolge (\textit{Kombinationen}) &
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$\lvert \widetilde{C}_N^{(K)} \rvert =
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\binom{N+K-1}{K} $ & $\lvert C_N^{(K)} \rvert
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= \binom{N}{K} $
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\end{tabular}
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\end{table}
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\item \pause Permutationen
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\begin{columns}
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\column{\kitfourcolumns}
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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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\Pi_N = \mleft\{ \mleft( a_1, \ldots, a_N
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\mright) \in \Omega : a_i \neq a_j, i \neq j
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\mright\}\\
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\begin{array}{r}
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\text{Alle Elemente von $\Omega$ unterscheidbar:} \\
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\text{Jeweils $L_1, L_2, \ldots, L_M$ der Elemente
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sind gleich:}
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\end{array}
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\hspace{5mm}
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\begin{array}{rl}
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\lvert \Pi_N \rvert &= N! \\
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\lvert \Pi_N^{(L_1,
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L_2, \ldots, L_M)} \rvert &=
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\frac{N!}{L_1!L_2!\cdots L_M!}
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\end{array}
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\end{gather*}
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\end{subfigure}%
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\begin{subfigure}[c]{0.4\textwidth}
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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=\textwidth,
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height=0.5\textwidth,
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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{subfigure}
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\end{figure}
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\column{\kittwocolumns}
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\begin{lightgrayhighlightbox}
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Beispiel:
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\begin{gather*}
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\Omega = {A, B, C}\\
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\Pi_N = \{ (A,B,C), (A,C,B), (B,A,C),\\
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(B,C,A), (C,A,B), (C,B,A)\}
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\end{gather*}
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\vspace*{-14mm}%
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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}
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\frametitle{Zusammenfassung}
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\begin{columns}
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\column{\kitthreecolumns}
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\begin{greenblock}{Potenzmenge}
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\vspace*{-6mm}
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\begin{gather*}
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\mathcal{P}\mleft( \Omega \mright) = \mleft\{ A:
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A \subseteq \Omega \mright\}
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\end{gather*}
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\end{greenblock}
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\column{\kitthreecolumns}
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\begin{greenblock}{Permutationen}
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\vspace*{-6mm}
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\begin{align*}
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\lvert \Pi_N \rvert &= N! \\
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\lvert \Pi_N^{(L_1, L_2, \ldots, L_M)} \rvert &=
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\frac{N!}{L_1!L_2!\cdots L_M!}
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\end{align*}
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\end{greenblock}
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\end{columns}
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\begin{columns}
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\column{\kitonecolumn}
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\column{\kitfourcolumns}
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\begin{greenblock}{Variationen \& Kombinationen }
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\begin{table}
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\begin{tabular}{r||l|l}
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& Mit Zurücklegen & Ohne Zurücklegen
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\\\hline\hline Mit Reihenfolge
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(\textit{Variationen}) & $\lvert
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\widetilde{V}_N^{(K)} \rvert = N^K$ & $\lvert
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V_N^{(K)}\rvert = \frac{N!}{(N-K)!} $ \\\hline
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Ohne Reihenfolge (\textit{Kombinationen}) &
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$\lvert \widetilde{C}_N^{(K)} \rvert =
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\binom{N+K-1}{K} $ & $\lvert C_N^{(K)} \rvert
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= \binom{N}{K} $
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\end{tabular}
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\end{table}
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\end{greenblock}
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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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% 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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@ -367,48 +453,72 @@
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\end{frame}
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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
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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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% tex-fmt: off
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\begin{enumerate}[a{)}]
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\item Die Ergebnismenge sei $\Omega = \{S, K, T, P\}$. Wie lautet die
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Potenzmenge $P(\Omega)$?\pause
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\begin{align*}
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\mathcal{P}(\Omega) = \{ &\emptyset, \mleft\{ S \mright\}, \mleft\{ K \mright\}, \mleft\{ T \mright\}, \mleft\{ P \mright\},\\
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&\mleft\{ S, K \mright\}, \mleft\{ S, T \mright\}, \mleft\{ S, P \mright\}, \mleft\{ K, T \mright\}, \mleft\{ K,P \mright\}, \mleft\{ T, P \mright\}, \\
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&\mleft\{ S, K, T \mright\}, \mleft\{ S, K, P \mright\}, \mleft\{ S, T, P \mright\}, \mleft\{ K, T, P \mright\}, \mleft\{ S, K, T, P \mright\}\}
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\end{align*}%
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\item \pause Für einen normalen Burger werden 3 der 4 möglichen Zutaten
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ausgewählt und in einer
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bestimmten Reihenfolge auf das Burgerbrötchen gelegt. Wie viele
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verschiedene normale
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Burger gibt es?\pause
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\begin{gather*}
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\lvert V_N^{(K)} \rvert = \frac{4!}{1!} = 24
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\end{gather*}
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\end{enumerate}
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% tex-fmt: on
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\end{frame}
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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
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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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% tex-fmt: off
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\begin{enumerate}[a{)}]
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\setcounter{enumi}{2}
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\item Ein Burger ``Spezial'' besteht ebenfalls aus 3 Zutaten. Jedoch
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können Tomate und Salat
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doppelt vorkommen. Wie viele verschiedene Burger „Spezial“ gibt es?\pause
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\begin{align*}
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n_\text{Burger} &= n_\text{Burger,alle Unterschiedlich} + n_{\text{Burger,2}\times\text{Salat}} + n_{\text{Burger,2}\times\text{Tomate}} \\
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&= 24 + 3\cdot 3 + 3\cdot 3 = 42
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\end{align*}
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\item \pause Der Burger „Jumbo“ enthält die folgende Menge an Zutaten: $\{S, S,
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T, T, K, K, K, P, P, P\}$
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die alle verwendet werden. Wie viele mögliche Belegungen des Burgers
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``Jumbo'' gibt es?\pause
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\begin{gather*}
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\lvert \Pi_N^{L_1,L_2,L_3,L_4} \rvert = \frac{10!}{2!2!3!3!} = 25200
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\end{gather*}
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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{Zusammenfassung}
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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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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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\end{gather*}
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\begin{figure}
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\centering
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\begin{subfigure}[c]{0.5\textwidth}
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\centering
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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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\end{gather*}
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\end{subfigure}%
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\begin{subfigure}[c]{0.4\textwidth}
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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=\textwidth,
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height=0.5\textwidth,
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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{subfigure}
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\end{figure}
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\end{frame}
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% TODO: Do we even want this section?
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\end{document}
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