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\begin{document} \punkte[21] @@ -33,7 +34,71 @@ \end{enumerate} \end{aufgabe} -\stepcounter{aufgabe} +\begin{aufgabe} + Ist $X$ ein Ereignis, so bezeichne $\overline{X}$ das Gegenereignis zu $X$. + Wir definieren $R, V$ und $S$ wie im Hinweis. Nach Aufgabenstellung gilt $\P(R) = \frac{1}{2},\P(V|R) = \frac{2}{3}$ und $\P(V|\overline R) = \frac{2}{3}$. Sei $A$ das Ereignis, dass Regen vorhergesagt wird. + $A$ tritt genau dann ein, wenn es regnet und die Wettervorhersage recht hat oder wenn es nicht regnet und die Wettervorhersage falsch liegt. Es gilt daher + $A = R \cap V \cup \overline R \cap \overline V$ und nach den De Morganschen Regeln + \[ + \overline{A} = \overline{R \cap V} \cap \overline{\overline{R} \cap \overline{V}} = (\overline{R} \cup \overline{V}) \cap (R \cup V) = \overline{R} \cap V \cup \overline{V} \cap R + \] + Weiter gilt $\P(S|A) = 1$ und $\P(S|\overline A) = \frac{1}{3}$. + %Die Wahrscheinlichkeit, dass Mr. Pickwick einen Schirm mitnimmt, obwohl kein Regen vorhergesagt wurde, ist unabhängig davon, ob es dann tatsächlich regnet oder nicht stets $\frac{1}{3}$. + %Daher sind die Ereignisse $S \cap \overline{A}$ und $R$ stochastisch unabhängig. + Da Mr. Pickwick nicht weiß, ob es regnen wird oder nicht, gilt sogar $\P(S|A\cap R) = \P(S|A\cap \overline R) = 1 $ und $\P(S|\overline A \cap R) = \P(S| \overline A \cap \overline R)$. + Daraus erhalten wir + \begin{align*} + 1 = \P(S|A\cap R) = \frac{\P(S \cap A \cap R)}{\P(A \cap R)} &\implies \P(S \cap A \cap R) = \P(A \cap R),\\ + 1 = \P(S|A\cap \overline R) = \frac{\P(S \cap A \cap \overline R)}{\P(A \cap \overline R)} &\implies \P(S \cap A \cap \overline R) = \P(A \cap \overline R),\\ + \frac{1}{3} = \P(S|\overline A\cap R) = \frac{\P(S \cap \overline A \cap R)}{\P(\overline A \cap R)} &\implies \P(S \cap \overline A \cap R) = \frac{1}{3}\P(\overline A \cap R),\\ + \frac{1}{3} = \P(S|\overline A\cap \overline R) = \frac{\P(S \cap \overline A \cap \overline R)}{\P(\overline A \cap \overline R)} &\implies \P(S \cap \overline A \cap \overline R) = \frac{1}{3}\P(\overline A \cap \overline R),\\ + \end{align*} + Zudem gilt + \begin{align*} + \frac{2}{3} = \P(V | R) = \frac{\P(V \cap R)}{\P(R)} = 2 P(V\cap R) &\implies P(V\cap R) = \frac{1}{3}\\ + \frac{2}{3} = \P(V | \overline R) = \frac{\P(V \cap \overline R)}{\P(\overline R)} = 2 P(V\cap \overline R) &\implies P(V\cap \overline R) = \frac{1}{3}\\ + \end{align*} + Daraus erhalten wir wegen $\P(X \cap Y + \P(X \cap \overline{Y}) = \P(X)$ sofort + \begin{align*} + \P(\overline V\cap R) &= \P(R) - \P(V\cap R) = \frac{1}{2} - \frac{1}{3} = \frac{1}{6}\\ + \P(\overline{V} \cap \overline{R}) &= \P(\overline{R}) - \P(V \cap \overline{R}) = \frac{1}{2} - \frac{1}{3} = \frac{1}{6} + \end{align*} + \begin{enumerate}[(a)] + \item Gesucht ist $\P(\overline{S}|R)$. + Wir berechnen zunächst + \begin{align*} + \P(S \cap R) &= \P(S \cap R \cap A) + \P(S \cap R\cap \overline{A})\\ + &= \P(S\cap A \cap R) + \P(S \cap \overline{A} \cap R)\\ + &= \P(A \cap R) + \frac{1}{3} \P(\overline{A} \cap R)\\ + &= \P((R \cap V \cup \overline R \cap \overline V) \cap R) + \frac{1}{3}\P((\overline{R} \cap V \cup \overline{V} \cap R) \cap R)\\ + &= \P(R\cap V) + \frac{1}{3} \P(R \cap \overline{V})\\ + &= \frac{1}{3} + \frac{1}{3} \frac{1}{6} = \frac{7}{18} + \end{align*} + Es gilt daher + \begin{align*} + \P(\overline{S}|R) &= \frac{\P(\overline{S}\cap R)}{\P(R)}\\ + &= 2 \cdot (\P(R) - \P(S \cap R))\\ + &= 1 - 2 \P(S \cap R)\\ + &= 1 - 2 \frac{7}{18} = \frac{18}{18} - \frac{14}{18} = \frac{2}{9} + \end{align*} + \item Gesucht ist $\P(S| \overline{R})$. + Wir berechnen zunächst + \begin{align*} + \P(S \cap \overline R) &= \P(S \cap \overline R \cap A) + \P(S \cap \overline R\cap \overline{A})\\ + &= \P(S\cap A \cap \overline R) + \P(S \cap \overline{A} \cap \overline R)\\ + &= \P(A \cap \overline R) + \frac{1}{3} \P(\overline{A} \cap \overline R)\\ + &= \P((R \cap V \cup \overline R \cap \overline V) \cap \overline R) + \frac{1}{3}\P((\overline{R} \cap V \cup \overline{V} \cap R) \cap \overline R)\\ + &= \P(\overline R\cap \overline V) + \frac{1}{3} \P(\overline R \cap V)\\ + &= \frac{1}{6} + \frac{1}{3} \frac{1}{3} = \frac{5}{18} + \end{align*} + Es gilt daher + \begin{align*} + \P(S|\overline R) &= \frac{\P(S\cap \overline R)}{\P(\overline{R})}\\ + &= 2 \cdot \P(S \cap \overline R)\\ + &= 2 \frac{5}{18} = \frac{10}{18} = \frac{5}{9} + \end{align*} + \end{enumerate} +\end{aufgabe} \begin{aufgabe} \begin{enumerate}[(a)] @@ -70,4 +135,14 @@ \end{enumerate} \end{aufgabe} +\begin{aufgabe} + %Sei $N$ die Anzahl der Zeichen in Goethes Faust und $M$ die Anzahl der verschiedenen Zeichen in Goethes Faust. + %Sei dann $A_i$ das Ereignis, dass es ein $j \leq i$ gibt, sodass $\forall 1\leq k \leq N$ der $j + k$-te Buchstabe, den der Affe tippt, genau dem $i$-ten Buchstaben von Goethes Faust entspricht. + %Die Weisheit entspricht dann wegen $A_i \subset A_j \forall i \leq j$ genau der Aussage + %\[ + % \forall m \in \N\colon \exists n\in \N\colon \P(A_n) = 1 \Leftrightarrow \P(\limsup\limits_{n \to \infty} A_n) + %\] + %Offensichtlich sind alle Ereignisse stochastisch unabhängig. Außerdem ist $\P(A_i) = \frac{1}{M^N} \forall i \in \N$. +\end{aufgabe} + \end{document}
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