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f-<(qyS0iUvPv?(6%EZjZ{K1GwPA;Y(4);F*6d_my diff --git a/material/ab1.tex b/material/ab1.tex index fe35902..dabb0e7 100644 --- a/material/ab1.tex +++ b/material/ab1.tex @@ -201,7 +201,7 @@ \newcommand{\curdate}{09.07.2026} -\newcommand{\curtitle}{Pascalsches Dreieck} +\newcommand{\curtitle}{Galton X Pascal} \title{\curtitle} \author{Fierke, Janik, Weidlich} @@ -215,13 +215,33 @@ \ifoot{\normalfont\normalsize Fierke, Janik, Weidlich} \section{\curtitle} - Im Stochastikunterricht ist dir sicherlich schoneinmal die Formel des Binomialkoeffienten begegnet: - \begin{definitionbox}[Binomialkoeffizient] - Der Binomialkoeffizient gibt die Anzahl der Möglichkeiten an, aus einer Menge mit $n$ Elementen $k$ verschiedene Elemente auszuwählen: - \[ + \begin{figure}[H] + \centering + % Erstes Bild + \begin{minipage}[t]{0.7\textwidth} + \centering + Im Stochastik Unterricht ist dir sicherlich schon einmal die + Formel des Binomialkoeffizienten (Abb. \ref{fig:bino}) begegnet. + \begin{definitionbox}[Binomialkoeffizient] + Der Binomialkoeffizient gibt die Anzahl der Möglichkeiten an, aus einer Menge mit $n$ Elementen $k$ verschiedene Elemente auszuwählen. + \end{definitionbox} + \end{minipage} + \hfill + % Zweites Bild + \begin{minipage}[t]{0.25\textwidth} + \centering + \[ \binom{n}{k}=\frac{n!}{k!(n-k)!} - \] - \end{definitionbox} + \] + \caption{Formel Binomialkoeffizienten} + \label{fig:bino} + \end{minipage} + \end{figure} + + + + Im Stochastikunterricht ist dir sicherlich schoneinmal die Formel des Binomialkoeffienten begegnet: + Zum Warmwerden berechnen wir: \begin{multicols}{3} @@ -233,7 +253,6 @@ $\binom{5}{5}=$ \hrulefill\\\\ $\binom{4}{2}=$ \hrulefill\\\\ - $\binom{n}{0}=$ \hrulefill { }für $n\in\mathbb{N}$\\\\ $\binom{n}{n}=$ \hrulefill { }für $n\in\mathbb{N}$\\\\ @@ -385,11 +404,11 @@ \filldraw[black] (\coordx, -\y) circle (2.5pt); % Erste fünf Nägel mit a,b,c,d,e beschriften - \stepcounter{labelcount} - \ifnum\value{labelcount}<11 - \node[anchor=south east] at (\coordx-0.16,-\y-0.3) - {\alph{labelcount}}; - \fi + %\stepcounter{labelcount} + %\ifnum\value{labelcount}<11 + %\node[anchor=south east] at (\coordx-0.16,-\y-0.3) + %{\alph{labelcount}}; + %\fi \fi % Wege nach unten zeichnen (auch von der letzten Reihe aus nicht nötig, @@ -398,6 +417,10 @@ % \draw[gray, thin] (\coordx, -\y) -- (\coordx - 1, -\y - 1); % \draw[gray, thin] (\coordx, -\y) -- (\coordx + 1, -\y - 1); % \fi + + %Beispiel einzeichnen: left left left + \draw[gray, thin] (0, 0) -- node[above left] {L} (-1,- 1) -- node[above left] {L} (-2,-2) -- node[above left] {L} (-3,-3); + \draw[gray, thin] (-2, -2) -- node[above right] {R} (-1,-3); } } \end{tikzpicture} @@ -427,11 +450,11 @@ \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 + %\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, @@ -445,11 +468,12 @@ \end{tikzpicture} \end{multicols} - \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 für Fach 2: $A_2=\{(a,b,d),(a,b,e)\} \Rightarrow \text{Pfade}_2=|A_2|=2$ + \item Zähle nun die verschiedenen Pfade, die eine Murmel nehmen kann, sodass sie in den darunterliegenden Fächern landet. + \scriptsize(Hinweis: Nutze die Beschreibung, dass eine Kugel immer nur nach rechts (r) oder links (l) fallen kann und schreibe das in Mengenschreibweise, sodass du keinen Pfad vergisst.) \normalsize \\ + Bsp: Fach 0: $A_0=\{(L,L,L)\} \Rightarrow \text{Pfade}_0=|A_0|=2$ \vspace{85pt} - \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 Dreieck. Was fällt dir auf? + \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 Dreieck. Was fällt dir auf? \vspace{85pt} \item Gib die Verteilung des Galtonbretts mithilfe einer Zufallsvariabel $X$ an. Wie könnte diese mit dem pascalschen Dreieck zusammenhängen?