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\begin{enumerate}[(i)] + \item Es ist für $A \in \mathcal{A}$: + $\mathbb{P}(A | B) = \frac{\mathbb{P}(A|B)}{\mathbb{P}(B)} \ge 0$, + da $\mathbb{P}$ W'maß. + \item $\mathbb{P}(\Omega | B) = \frac{\mathbb{P}(\Omega \cap B)}{\mathbb{P}(B)} + = \frac{\mathbb{P}(B)}{\mathbb{P}(B)} = 1$. + \item Seien $A_i \in \mathcal{A}$ mit $A_i$ paarweise disjunkt. Dann folgt + \begin{salign*} + \mathbb{P}\left( \bigcupdot_{i \in \N} A_i \right) + &= \frac{\mathbb{P}\left( \left( \bigcupdot_{i \in \N} A_i \right) \cap B \right) }{\mathbb{P}(B)} \\ + &= \frac{\mathbb{P}\left( \bigcupdot_{i \in \N} (A_i \cap B) \right) }{\mathbb{P}(B)} \\ + &= \sum_{i \in \N} \frac{\mathbb{P}(A_i \cap B)}{\mathbb{P}(B)} \\ + &= \sum_{i \in \N} \mathbb{P}(A_i|B) + ,\end{salign*} + wobei im 3. Schritt die $\sigma$-Additivität von $\mathbb{P}$ ausgenutzt wurde. + \end{enumerate} + \end{proof} + \item Es ist beispielsweise mit + $A = \emptyset\colon \mathbb{P}(\emptyset | \Omega) = \frac{\mathbb{P}(\emptyset \cap \Omega)}{\mathbb{P}(\Omega)} = \frac{0}{1} = 0 \neq 1$. + \end{enumerate} +\end{aufgabe} + +\stepcounter{aufgabe} + +\begin{aufgabe} + \begin{enumerate}[(a)] + \item Sei $A \in \mathcal{A}$. Dann ist $\mathbb{P}(\Omega \cap A) = \mathbb{P}(A) = 1 \cdot \mathbb{P}(A) = \mathbb{P}(\Omega) \mathbb{P}(A)$. Außerdem + gilt $\mathbb{P}(\emptyset \cap A) = \mathbb{P}(\emptyset) = 0 = 0 \cdot \mathbb{P}(A) = + \mathbb{P}(\emptyset) \mathbb{P}(A)$. + \item Seien $A, B, C$ gemeinsam stochastisch unabhängig. Dann ist + $\mathbb{P}((A \cap B) \cap C) = \mathbb{P}(A \cap B \cap C) + = \mathbb{P}(A) \mathbb{P}(B) \mathbb{P}(C) = \mathbb{P}(A \cap B) \mathbb{P}(C)$. + Außerdem gilt + $\mathbb{P}((A \cup B) \cap C) = \mathbb{P}((A \cap B) \cup (B \cap C)) + = \mathbb{P}(A \cap C) + \mathbb{P}(B \cap C) - \mathbb{P}(A \cap C) \cap (B \cap C) + = \mathbb{P}(A)\mathbb{P}(C) + \mathbb{P}(B) \mathbb{P}(C) - \mathbb{P}(A \cap B \cap C) + = \mathbb{P}(C) (\mathbb{P}(A) + \mathbb{P}(B) - \mathbb{P}(A \cap B)) + = \mathbb{P}(C) \mathbb{P}(A \cup B)$. + \item Da der Würfel Laplace verteilt angenommen ist, folgt direkt + $\mathbb{P}(A) = \frac{1}{2}$ und $\mathbb{P}(B) = \frac{1}{2}$. Da + die Summe der Augenzahlen genau dann gerade ist, wenn einer der Würfe eine gerade + und einer der Würfe eine ungerade Zahl ergibt, folgt $\mathbb{P}(C) = \frac{1}{2}$. + Dabei gilt $\mathbb{P}(A \cap B \cap C) = 0$, da die Summe der Augenzahlen gerade ist, falls + beide Würfe gerade Augenzahlen ergeben. + + Die Ereignisse $A, B, C$ sind paarweise unabhängig, denn + \begin{salign*} + \mathbb{P}(A \cap C) &= \mathbb{P}(\text{,,1. Wurf gerade, 2. ungerade''}) = \frac{1}{4} + = \mathbb{P}(A) \mathbb{P}(C) \\ + \mathbb{P}(A \cap B) &= \mathbb{P}(\text{,,1. und 2. Wurf gerade''}) + = \frac{1}{4} = \mathbb{P}(A) \mathbb{P}(B) \\ + \mathbb{P}(B \cap C) &= \mathbb{P}(\text{,,1. Wurf ungerade, 2. gerade''}) + = \frac{1}{4} = \mathbb{P}(B) \mathbb{P}(C) + .\end{salign*} + + Aber $\mathbb{P}(A \cap B \cap C) = 0 \neq \frac{1}{8} = \mathbb{P}(A) \mathbb{P}(B) \mathbb{P}(C)$ + \end{enumerate} +\end{aufgabe} + +\end{document}