diff --git a/src/2025-11-21/presentation.tex b/src/2025-11-21/presentation.tex index b15d946..bf3e15e 100644 --- a/src/2025-11-21/presentation.tex +++ b/src/2025-11-21/presentation.tex @@ -43,7 +43,7 @@ \usepackage{tikz} \usepackage{tikz-3dplot} -\usetikzlibrary{spy, external, intersections} +\usetikzlibrary{spy, external, intersections, positioning} %\tikzexternalize[prefix=build/] \usepackage{pgfplots} @@ -80,6 +80,145 @@ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \subsection{Theorie Wiederholung} +\begin{frame} + \frametitle{Bedingte Wahrscheinlichkeiten \& Bayes} + + \vspace*{-10mm} + + \begin{columns} + \column{\kitthreecolumns} + \begin{itemize} + \item Definition der bedingten Wahrscheinlichkeit + \begin{gather*} + P(A\vert B) = \frac{P(AB)}{P(B)} + \end{gather*} + \item Formel von Bayes + \begin{gather*} + P(A\vert B) = \frac{P(B\vert A) P(A)}{P(B)} + \end{gather*} + \end{itemize} + \column{\kitthreecolumns} + \begin{figure} + \centering + \begin{tikzpicture} + \node[rectangle, minimum width=8cm, minimum height=5cm, + draw, line width=1pt, fill=black!20] at (0,0) {}; + \node [circle, minimum size = 4cm, + draw, line width=1pt, fill=KITgreen, + fill opacity = 0.5] at (1.25cm,0) {}; + \draw[line width=1pt, fill=KITblue, + fill opacity = 0.5, rounded corners=5mm] + (-2.4cm, -2.25cm) -- (-2.4cm, 2.25cm) -- (1.1cm,0) -- cycle; + + \node[left] at (4cm, 2cm) {\Large $\Omega$}; + \node at (-1.8cm, 0) {$A$}; + \node at (1.8cm, 0) {$B$}; + \node at (0, 0) {$AB$}; + \end{tikzpicture} + \end{figure} + \end{columns} + \vspace*{1cm} + \pause + \begin{columns} + \column{\kitthreecolumns} + \begin{itemize} + \item Satz der totalen Wahrscheinlichkeit + % tex-fmt: off + \begin{gather*} + \text{Voraussetzungen: }\hspace{5mm} \left\{ + \begin{array}{l} + A_1, A_2, \ldots \text{ disjunkt}\\ + \displaystyle\sum_{n} A_n = \Omega + \end{array} + \right.\\[1em] + P(B) = \sum_{n} P(B\vert A_n)P(A_n)\\ + \end{gather*} + % tex-fmt: on + \end{itemize} + \column{\kitthreecolumns} + \begin{figure} + \centering + \begin{tikzpicture} + \newcommand{\hordist}{1.2cm} + \newcommand{\vertdist}{2cm} + + \node[circle, fill=KITgreen, inner sep=0pt, + minimum size=3mm] (root) at (0, 0) {}; + \node[circle, fill=KITgreen, inner sep=0pt, + minimum size=3mm, below left=\vertdist and + 2.4*\hordist of root] (n1) {}; + \node[circle, fill=KITgreen, inner sep=0pt, + minimum size=3mm, below right=\vertdist and + 2.4*\hordist of root] (n2) {}; + \node[circle, fill=KITgreen, inner sep=0pt, + minimum size=3mm, below left=\vertdist and \hordist + of n1] (n11) {}; + \node[circle, fill=KITgreen, inner sep=0pt, + minimum size=3mm, below right=\vertdist and \hordist + of n1] (n12) {}; + \node[circle, fill=KITgreen, inner sep=0pt, + minimum size=3mm, below left=\vertdist and \hordist + of n2] (n21) {}; + \node[circle, fill=KITgreen, inner sep=0pt, + minimum size=3mm, below right=\vertdist and \hordist + of n2] (n22) {}; + + \draw[-{Latex}, line width=1pt] (root) -- (n1); + \draw[-{Latex}, line width=1pt] (root) -- (n2); + \draw[-{Latex}, line width=1pt] (n1) -- (n11); + \draw[-{Latex}, line width=1pt] (n1) -- (n12); + \draw[-{Latex}, line width=1pt] (n2) -- (n21); + \draw[-{Latex}, line width=1pt] (n2) -- (n22); + + \node[left] at ($(root)!0.4!(n1)$) {$P(A_1)$}; + \node[right] at ($(root)!0.4!(n2)$) {$P(A_2)$}; + + \node[left] at ($(n1)!0.4!(n11)$) {$P(B\vert A_1)$}; + \node[right] at ($(n1)!0.2!(n12)$) {$P(C\vert A_1)$}; + \node[left] at ($(n2)!0.6!(n21)$) {$P(B\vert A_2)$}; + \node[right] at ($(n2)!0.4!(n22)$) {$P(C\vert A_2)$}; + + \node[below] at (n11) {$P(BA_1)$}; + \node[below] at (n12) {$P(CA_2)$}; + \node[below] at (n21) {$P(BA_1)$}; + \node[below] at (n22) {$P(CA_2)$}; + \end{tikzpicture} + \end{figure} + \end{columns} +\end{frame} + +\begin{frame} + \frametitle{Zusammenfassung} + + \begin{columns} + \column{\kitthreecolumns} + \begin{greenblock}{Bedingte Wahrscheinlichkeit} + \vspace*{-6mm} + \begin{gather*} + P(A\vert B) = \frac{P(AB)}{P(B)} + \end{gather*} + \end{greenblock} + \column{\kitthreecolumns} + \begin{greenblock}{Formel von Bayes} + \vspace*{-6mm} + \begin{gather*} + P(A\vert B) = \frac{P(AB)}{P(B)} + \end{gather*} + \end{greenblock} + \end{columns} + \begin{columns} + \column{\kitonecolumn} + \column{\kitthreecolumns} + \begin{greenblock}{Satz der totalen Wahrscheinlichkeit} + \vspace*{-6mm} + \begin{gather*} + P(B) = \sum_{n} P(B\vert A_n)P(A_n) + \end{gather*} + \end{greenblock} + \column{\kitonecolumn} + \end{columns} +\end{frame} + % \begin{frame}{Ereignisse \& Laplace} % \vspace*{-15mm} % \begin{itemize} @@ -160,7 +299,8 @@ % \begin{columns} % \column{\kitthreecolumns} % \begin{gather*} -% P_r = \frac{\binom{R}{r}\binom{N-R}{n-r}}{\binom{N}{n}} +% P_r = +% \frac{\binom{R}{r}\binom{N-R}{n-r}}{\binom{N}{n}} % \end{gather*} % \column{\kitthreecolumns} % \begin{lightgrayhighlightbox} @@ -270,7 +410,8 @@ % \begin{align*} % \mathcal{P}(\Omega) = \{ &\emptyset, % \mleft\{ A \mright\}, \mleft\{ B \mright\}, -% \mleft\{ C \mright\}, \mleft\{ A, B \mright\},\\ +% \mleft\{ C \mright\}, \mleft\{ A, B +% \mright\},\\ % &\mleft\{ A, C \mright\}, % \mleft\{ B, C \mright\}, \mleft\{ A, B, C % \mright\} \} @@ -302,7 +443,8 @@ % \mright) \in \Omega : a_i \neq a_j, i \neq j % \mright\}\\ % \begin{array}{r} -% \text{Alle Elemente von $\Omega$ unterscheidbar:} \\ +% \text{Alle Elemente von $\Omega$ +% unterscheidbar:} \\ % \text{Jeweils $L_1, L_2, \ldots, L_M$ der Elemente % sind gleich:} % \end{array} @@ -382,7 +524,8 @@ Bei einer Qualitätskontrolle können Werkstücke zwei Fehler aufweisen: Fehler A, Fehler B, oder - beide Fehler gleichzeitig. Die folgenden Wahrscheinlichkeiten sind bekannt: + beide Fehler gleichzeitig. Die folgenden Wahrscheinlichkeiten + sind bekannt: \begin{itemize} \item mit Wahrscheinlichkeit 0,05 hat ein Werkstück den Fehler A \item mit Wahrscheinlichkeit 0,01 hat ein Werkstück beide Fehler