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q@F&w(b*L3)u=oFU4?U1p?nobZTRSQmh`1C?N|K6~S5;Sy>c0T=DM;V| diff --git a/material/ab1.tex b/material/ab1.tex index 6d921ea..8dec0b4 100644 --- a/material/ab1.tex +++ b/material/ab1.tex @@ -26,7 +26,9 @@ \usepackage{xcolor} \usepackage{setspace} \usepackage{letltxmacro} +\usepackage{wrapfig} +\usetikzlibrary{calc} \setlist{itemsep=0.3em, topsep=0.5em, parsep=0pt} @@ -339,5 +341,76 @@ Entwirf einen Computeralgorithmus, der den Binomialkoeffizienten berechnet, ohne dass dieser die Fakultät benutzt! \end{extrataskbox} + {\begin{wrapfigure}{r}{0.4\textwidth} + \centering + \includegraphics[width=0.7\linewidth]{galtonbrett} + + \caption{} + \label{fig:galtonbrett} + \end{wrapfigure} + + Beim Galtonbrett lassen wir kleine Murmeln auf dreieckig angeordnete Stäbchen fallen. Trifft eine Murmel auf ein Stäbchen, fällt sie gleichwahrscheinlich entweder nach links, oder nach rechts. + + \begin{enumerate} + \item Zeichne in das Galtonbrett alle Wege ein, die eine Murmel nehmen kann, wenn sie auf den obersten Stab fallen gelassen wird.\\ + \begin{tikzpicture}[scale=0.8] + % Definition der Reihen + \def\rows{5} + + % Zähler für die Beschriftung + \newcounter{labelcountzwo} + + % Fächer (Bins) am unteren Ende basierend auf der Reihenanzahl + \foreach \i in {1,...,\rows} { + \pgfmathsetmacro{\xpos}{\i * 2 - \rows} + \draw[thick] (\xpos - 0.4, -\rows - 1.5) rectangle (\xpos + 0.4, -\rows - 0.2); + \node at (\xpos, -\rows - 1.8) {\i}; + } + + % Dreieckiges Nagelbrett + \foreach \y in {1,...,\rows} { + \foreach \x in {1,...,\y} { + \pgfmathsetmacro{\coordx}{2 * \x - \y} + + % Nagel zeichnen + % Nagel nur bis zur vorletzten Reihe zeichnen + \ifnum \y < \rows + \filldraw[black] (\coordx, -\y) circle (2.5pt); + + % Erste fünf Nägel mit a,b,c,d,e beschriften + \stepcounter{labelcountzwo} + \ifnum\value{labelcountzwo}<11 + \node[anchor=south east] at (\coordx-0.16,-\y-0.3) + {\alph{labelcountzwo}}; + \fi + \fi + + % Wege nach unten zeichnen (auch von der letzten Reihe aus nicht nötig, + % daher weiterhin nur bis zur vorletzten Reihe) + \ifnum \y < \rows + \draw[gray, thin] (\coordx, -\y) -- (\coordx - 1, -\y - 1); + \draw[gray, thin] (\coordx, -\y) -- (\coordx + 1, -\y - 1); + \fi + } + } + \end{tikzpicture} + \end{enumerate}} + \begin{enumerate} + + \item Zähle nun die verschiedenen Pfade, die eine Mumel nehmen kann, sodass sie in den darunterliegenden Fächern landet. Beschrifte dafür die Pfade und nutze die Mengenschreibweise, um keinen Pfad zu vergessen.\\ + Beispiel: Fach 1: $A_1=\{(a,c,g)\} \Rightarrow \text{Pfade}_1=|A_1|=1$ + \vspace{100pt} + + \item Dir ist sicherlich schon aufgefallen, dass die meisten Pfade in die Mitte führen. Untersuche nun die Anzahl der Pfade zu jedem der Stäbchen und vergleiche diese mit dem pascalschen Dreicek. Was fällt dir auf? + \vspace{100pt} + + \item Gib die Verteilung des Galtonbretts mithilfe einer Zufallsvariabel $X$ an. Wie könnte diese mit dem pascalschen Dreieck zusammenhängen? + \vspace{100pt} + \end{enumerate} + + + + + \label{LastPage} \end{document}