Sofia wishes to go to the cinema. She can go by bike or by bus.

\section*{Part A: Using the bus}
We assume in this part that Sofia uses the bus to go to the cinema. The duration of the journey between her home and the cinema (expressed in minutes) is modelled by the random variable $T _ { B }$ which follows the uniform distribution on [12; 15].

\begin{enumerate}
  \item Prove that the probability that Sofia takes between 12 and 14 minutes is $\frac { 2 } { 3 }$.
  \item Give the average duration of the journey.
\end{enumerate}

\section*{Part B: Using her bike}
We now assume that Sofia chooses to use her bike.\\
The duration of the journey (expressed in minutes) is modelled by the random variable $T _ { v }$ which follows the normal distribution with mean $\mu = 14$ and standard deviation $\sigma = 1,5$.

\begin{enumerate}
  \item What is the probability that Sofia takes less than 14 minutes to go to the cinema? What is the probability that Sofia takes between 12 and 14 minutes to go to the cinema? Round the result to $10 ^ { - 3 }$.
\end{enumerate}

\section*{Part C: Playing with dice}
Sofia is hesitating between the bus and the bike. She decides to roll a fair 6-sided die.\\
If she gets 1 or 2, she takes the bus, otherwise she takes her bike. We denote:
\begin{itemize}
  \item $B$ the event ``Sofia takes the bus'';
  \item $V$ the event ``Sofia takes her bike'';
  \item C the event ``Sofia takes between 12 and 14 minutes to go to the cinema''.
\end{itemize}

\begin{enumerate}
  \item Prove that the probability, rounded to $10 ^ { - 2 }$, that Sofia takes between 12 and 14 minutes is 0.49.
  \item Given that Sofia took between 12 and 14 minutes to go to the cinema, what is the probability, rounded to $10 ^ { - 2 }$, that she used the bus?
\end{enumerate}