tut3: Add examples of distributions
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@ -77,6 +77,9 @@
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\section{Aufgabe 1}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\subsection{Theorie Wiederholung}
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\begin{frame}
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\frametitle{Zufallsvariablen \& Verteilungen}
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@ -202,7 +205,8 @@
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\vspace*{-10mm}
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\end{lightgrayhighlightbox}
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\end{columns}
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\pause \item Einige Kenngrößen von Verteilungen
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\pause
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\item Einige Kenngrößen von Verteilungen
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\vspace*{2mm}
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\begin{columns}[t]
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\column{\kittwocolumns}
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@ -245,116 +249,74 @@
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\end{itemize}
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\end{frame}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\subsection{Theorie Wiederholung}
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\begin{frame}
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\frametitle{Beispiele von Verteilungen}
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\vspace*{-18mm}
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\begin{columns}[t]
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\column{\kittwocolumns}
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\centering
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\textbf{Bernoulli Verteilung}\\
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\vspace*{10mm}
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$X$ kann nur die Werte $0$ oder $1$\\ annehmen
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\rule{0.9\textwidth}{0.4pt}
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\begin{gather*}
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X \sim \text{Bernoulli}(p)
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\end{gather*}
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\begin{gather*}
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P(X=0) = 1-p, \hspace{5mm} P(X=1) = p
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\end{gather*}
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\begin{align*}
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E(X) &= p\\
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V(X) &= p(1-p)
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\end{align*}
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\column{\kittwocolumns}
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\centering
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\textbf{Binomialverteilung}\\
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\vspace*{10mm}
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$X\equiv$ ``Zählen der Treffer bei $N$ unabhängigen Versuchen''
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\rule{0.9\textwidth}{0.4pt}
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\begin{gather*}
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X \sim \text{Bin}(N,p)
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\end{gather*}
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\begin{gather*}
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P_X(k) = \binom{N}{k} p^k (1-p)^{1-k}
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\end{gather*}
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\begin{align*}
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E(X) &= Np\\
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V(X) &= Np(1-p)
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\end{align*}
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\column{\kittwocolumns}
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\centering
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\textbf{Poisson Verteilung}\\
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\vspace*{10mm}
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Binomialverteilung für $N\rightarrow \infty$ mit $pN=\text{const.}=: \lambda$
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\rule{0.9\textwidth}{0.4pt}
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\begin{gather*}
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X \sim \text{Poisson}(\lambda)
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\end{gather*}
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\begin{gather*}
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P_X(k) = \frac{\lambda^k}{k!}e^{-\lambda}
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\end{gather*}
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\begin{align*}
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E(X) &= \lambda\\
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V(X) &= \lambda
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\end{align*}
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\end{columns}
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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}{Binomialverteilung}
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adsf
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\end{greenblock}
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\end{columns}
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\end{frame}
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% \begin{frame}
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% \frametitle{Bedingte Wahrscheinlichkeiten \& Bayes}
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%
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% \vspace*{-10mm}
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%
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% \begin{columns}
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% \column{\kitthreecolumns}
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% \begin{itemize}
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% \item Definition der bedingten Wahrscheinlichkeit
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% \begin{gather*}
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% P(A\vert B) = \frac{P(AB)}{P(B)}
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% \end{gather*}
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% \item Formel von Bayes
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% \begin{gather*}
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% P(A\vert B) = \frac{P(B\vert A) P(A)}{P(B)}
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% \end{gather*}
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% \end{itemize}
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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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% \node[rectangle, minimum width=8cm, minimum height=5cm,
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% draw, line width=1pt, fill=black!20] at (0,0) {};
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% \node [circle, minimum size = 4cm,
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% draw, line width=1pt, fill=KITgreen,
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% fill opacity = 0.5] at (1.25cm,0) {};
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% \draw[line width=1pt, fill=KITblue,
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% fill opacity = 0.5, rounded corners=5mm]
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% (-2.4cm, -2.25cm) -- (-2.4cm, 2.25cm) -- (1.1cm,0) -- cycle;
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%
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% \node[left] at (4cm, 2cm) {\Large $\Omega$};
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% \node at (-1.8cm, 0) {$A$};
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% \node at (1.8cm, 0) {$B$};
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% \node at (0, 0) {$AB$};
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% \end{tikzpicture}
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% \end{figure}
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% \end{columns}
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% \vspace*{1cm}
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% \pause
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% \begin{columns}
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% \column{\kitthreecolumns}
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% \begin{itemize}
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% \item Satz der totalen Wahrscheinlichkeit
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% % tex-fmt: off
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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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% A_1, A_2, \ldots \text{ disjunkt}\\
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% \displaystyle\sum_{n} A_n = \Omega
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% \end{array}
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% \right.\\[1em]
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% P(B) = \sum_{n} P(B\vert A_n)P(A_n)\\
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% \end{gather*}
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% % tex-fmt: on
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% \end{itemize}
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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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% \newcommand{\hordist}{1.2cm}
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% \newcommand{\vertdist}{2cm}
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%
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% \node[circle, fill=KITgreen, inner sep=0pt,
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% minimum size=3mm] (root) at (0, 0) {};
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% \node[circle, fill=KITgreen, inner sep=0pt,
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% minimum size=3mm, below left=\vertdist and
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% 2.4*\hordist of root] (n1) {};
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% \node[circle, fill=KITgreen, inner sep=0pt,
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% minimum size=3mm, below right=\vertdist and
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% 2.4*\hordist of root] (n2) {};
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% \node[circle, fill=KITgreen, inner sep=0pt,
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% minimum size=3mm, below left=\vertdist and \hordist
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% of n1] (n11) {};
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% \node[circle, fill=KITgreen, inner sep=0pt,
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% minimum size=3mm, below right=\vertdist and \hordist
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% of n1] (n12) {};
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% \node[circle, fill=KITgreen, inner sep=0pt,
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% minimum size=3mm, below left=\vertdist and \hordist
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% of n2] (n21) {};
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% \node[circle, fill=KITgreen, inner sep=0pt,
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% minimum size=3mm, below right=\vertdist and \hordist
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% of n2] (n22) {};
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%
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% \draw[-{Latex}, line width=1pt] (root) -- (n1);
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% \draw[-{Latex}, line width=1pt] (root) -- (n2);
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% \draw[-{Latex}, line width=1pt] (n1) -- (n11);
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% \draw[-{Latex}, line width=1pt] (n1) -- (n12);
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% \draw[-{Latex}, line width=1pt] (n2) -- (n21);
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% \draw[-{Latex}, line width=1pt] (n2) -- (n22);
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%
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% \node[left] at ($(root)!0.4!(n1)$) {$P(A_1)$};
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% \node[right] at ($(root)!0.4!(n2)$) {$P(A_2)$};
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%
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% \node[left] at ($(n1)!0.4!(n11)$) {$P(B\vert A_1)$};
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% \node[right] at ($(n1)!0.2!(n12)$) {$P(C\vert A_1)$};
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% \node[left] at ($(n2)!0.6!(n21)$) {$P(B\vert A_2)$};
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% \node[right] at ($(n2)!0.4!(n22)$) {$P(C\vert A_2)$};
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%
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% \node[below] at (n11) {$P(BA_1)$};
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% \node[below] at (n12) {$P(CA_2)$};
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% \node[below] at (n21) {$P(BA_1)$};
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% \node[below] at (n22) {$P(CA_2)$};
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% \end{tikzpicture}
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% \end{figure}
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% \end{columns}
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% \end{frame}
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%
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% \begin{frame}
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% \frametitle{Zusammenfassung}
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%
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@ -691,8 +653,7 @@
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% beobachtet. Der Fehler tritt
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% mit der Wahrscheinlichkeit $0,01$ ein, wenn weder Fehler $A$ noch $B$
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% eingetreten sind und mit der
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% Wahrscheinlichkeit $0,02$, wenn sowohl Fehler $A$ als auch
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$B$ eingetreten
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% Wahrscheinlichkeit $0,02$, wenn sowohl Fehler $A$ als auch $B$ eingetreten
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% sind. In allen anderen
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% Fällen tritt der Fehler $C$ nicht auf.
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%
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@ -756,8 +717,7 @@ $B$ eingetreten
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% beobachtet. Der Fehler tritt
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% mit der Wahrscheinlichkeit $0,01$ ein, wenn weder Fehler $A$ noch $B$
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% eingetreten sind und mit der
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% Wahrscheinlichkeit $0,02$, wenn sowohl Fehler $A$ als auch
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$B$ eingetreten
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% Wahrscheinlichkeit $0,02$, wenn sowohl Fehler $A$ als auch $B$ eingetreten
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% sind. In allen anderen
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% Fällen tritt der Fehler $C$ nicht auf.
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%
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