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\hrule
\begin{center}
{\Large \textbf{Information Theory SS 2025}} \\[1ex]
\end{center}
\begin{center}
\textbf{\large {Institut für Diskrete Mathematik (5050), TU Graz}}
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\begin{center}
\textbf{\large Exercise sheet \thesection\ (13.03.2025)} \\[1ex]
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\begin{bsp}
A coin with possible outcomes 0 (head) and 1 (tail) is thrown 3 times $\left(\Omega=\{0,1\}^3\right)$. What is the minimal $\sigma$-algebra containing
\begin{enumerate}
\item the event [the outcome of the first throw is 1]?
\item the event [the sum of the outcomes is even]?
\item the event [the outcome of the first throw is 1 and the sum of the outcomes is even]?
\item the two events [the outcome of the first throw is 1] and [the sum of the outcomes is even]?
\end{enumerate}
\end{bsp} 

\begin{bsp}
Consider two independent rolls of a fair six-sided die. Let $i$ be the outcome of the first roll and $j$ the outcome of the second roll. Consider the following events:
\begin{equation*}
    A=[i \text{ is even}], \quad B=[i \text{ divides } 6], \quad C=[i+j=10]. 
\end{equation*}
\begin{enumerate}
    \item Show that $\mathbb{P}(A \cap B \cap C) = \mathbb{P}(A) \, \mathbb{P}(B) \, \mathbb{P}(C)$.
    \item Are the three events independent?
\end{enumerate}
\end{bsp}

\begin{bsp}
We are given two urns: urn 1 contains 8 black balls and 2 white balls and urn 2 contains 3 black balls and 9 white balls. First we draw randomly 2 balls from urn 1 and put them in urn 2. Then we draw randomly a ball from urn 2.
\begin{enumerate}
    \item Compute the probability that the drawn balls from urn 1 are both black.
    \item Compute the probability that the drawn balls from urn 1 are both black when given that the ball drawn from urn 2 is black. 
\end{enumerate}
\end{bsp}


\begin{bsp}
Let $(\Omega, \mathcal{A},\mathbb{P})$ be a probability space and $A_1, A_2, \dots, A_n \in  \mathcal{A}$ events. Show that
\begin{enumerate}
\item $\displaystyle \mathbb{P}\left(\bigcup_{i=1}^n A_i \right)= \sum_{k=1}^n \left( (-1)^{k-1} \sum_{\substack{I \subseteq \{1,\dots, n\} \\ |I|=k}} \mathbb{P}\left( \bigcap_{i \in I} A_i \right)\right).$
\item If the $A_i$ are disjoint and $\bigcup_{i=1}^n A_i = \Omega$, then for every $B \in \mathcal{A}$
\[\mathbb{P}(B) = \sum_{i=1}^n \mathbb{P}(B \mid A_i) \mathbb{P}(A_i).\]
\end{enumerate}
\end{bsp}

\begin{bsp}
Let $\Omega$ be the set of all finite sequences of zeros and ones.
\begin{enumerate}
	\item Find a probability measure $\mathbb{P}$ on $\Ac = \Pc (\Omega)$, such that for each finite sequence $\omega \in \Omega$, $\mathbb{P}(\{ \omega \})>0$.
 \item Is $\Omega$ a countable set? 
\end{enumerate}

(A \emph{countable} set is a set which can be \emph{enumerated} $\Omega = \{ \omega_n \colon n \in \mathbb{N}\})$
\end{bsp} 

\begin{bsp}
Let $\Omega$ be a sample set and let $(\Ac_n)_{n \in \mathbb{N}}$ with $\Ac_n \subseteq \Pc(\Omega)$ be a family of $\sigma$-algebras.
Show that: 
\begin{enumerate}
 \item If $\Ac_{n+1} \subseteq \Ac_n$ for every $n$ then $\bigcap_{n} \Ac_n$ is a $\sigma$-algebra.
 \item If $\Ac_{n} \subseteq \Ac_{n+1}$ for every $n$ then $\bigcup_{n} \Ac_n$ is, in general, not a $\sigma$-algebra. 
\end{enumerate}
\end{bsp} 

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