Exercise 4 For each of the following statements, indicate whether it is true or false. Each answer must be justified. An unjustified answer earns no points. A museum offers visits with or without an audioguide. Tickets can be purchased online or directly at the counter.
When a person buys their ticket online, a validation code is sent to them by SMS so they can confirm their purchase. This code is generated randomly and consists of 4 digits that are pairwise distinct, with the first digit being different from 0. Statement 1: The number of different codes that can be generated is 5040.
A study made it possible to consider that:
the probability that a person chooses the audioguide given that they bought their ticket online is equal to 0{,}8;
the probability that a person buys their ticket online is equal to 0{,}7;
the probability that a person opts for a visit without an audioguide is equal to 0{,}32.
Statement 2: The probability that a visitor does not take the audioguide given that they bought their ticket at the counter is greater than two thirds.
We randomly choose 12 visitors to this museum. We assume that the choice of the ``audioguide'' option is independent from one visitor to another. Statement 3: The probability that exactly half of these visitors opt for the audioguide is equal to $924 \times 0{,}2176^6$.
When a person has an audioguide, they can choose from three routes:
a first one lasting fifty minutes,
a second one lasting one hour and twenty minutes,
a third one lasting one hour and forty minutes.
The tour time can be modelled by a random variable $X$ whose probability distribution is given below:
$x_i$
$50\,\min$
$1\,\mathrm{h}\,20\,\min$
$1\,\mathrm{h}\,40\,\min$
$P(X = x_i)$
0{,}1
0{,}6
0{,}3
Statement 4: The expectation of $X$ is 77 minutes.
\textbf{Exercise 4}
For each of the following statements, indicate whether it is true or false.\\
Each answer must be justified.\\
An unjustified answer earns no points.\\
A museum offers visits with or without an audioguide. Tickets can be purchased online or directly at the counter.
\begin{enumerate}
\item When a person buys their ticket online, a validation code is sent to them by SMS so they can confirm their purchase.\\
This code is generated randomly and consists of 4 digits that are pairwise distinct, with the first digit being different from 0.
\textbf{Statement 1:} The number of different codes that can be generated is 5040.
\item A study made it possible to consider that:
\begin{itemize}
\item the probability that a person chooses the audioguide given that they bought their ticket online is equal to 0{,}8;
\item the probability that a person buys their ticket online is equal to 0{,}7;
\item the probability that a person opts for a visit without an audioguide is equal to 0{,}32.
\end{itemize}
\textbf{Statement 2:} The probability that a visitor does not take the audioguide given that they bought their ticket at the counter is greater than two thirds.
\item We randomly choose 12 visitors to this museum.
We assume that the choice of the ``audioguide'' option is independent from one visitor to another.
\textbf{Statement 3:} The probability that exactly half of these visitors opt for the audioguide is equal to $924 \times 0{,}2176^6$.
\item When a person has an audioguide, they can choose from three routes:
\begin{itemize}
\item a first one lasting fifty minutes,
\item a second one lasting one hour and twenty minutes,
\item a third one lasting one hour and forty minutes.
\end{itemize}
The tour time can be modelled by a random variable $X$ whose probability distribution is given below:
\begin{center}
\begin{tabular}{|c|c|c|c|}
\hline
$x_i$ & $50\,\min$ & $1\,\mathrm{h}\,20\,\min$ & $1\,\mathrm{h}\,40\,\min$ \\
\hline
$P(X = x_i)$ & 0{,}1 & 0{,}6 & 0{,}3 \\
\hline
\end{tabular}
\end{center}
\textbf{Statement 4:} The expectation of $X$ is 77 minutes.
\end{enumerate}