For each of the following five statements, indicate whether it is true or false and justify the answer. An unjustified answer is not taken into account. An absence of answer is not penalized.
Zoé goes to work on foot or by car. Where she lives, it rains one day out of four. When it rains, Zoé goes to work by car in $80\%$ of cases. When it does not rain, she goes to work on foot with a probability equal to 0.6. Statement $\mathbf{n^o 1}$: ``Zoé uses the car one day out of two.''
In the set $E$ of outcomes of a random experiment, we consider two events $A$ and $B$. Statement $\mathbf{n^o 2}$: ``If $A$ and $B$ are independent, then $A$ and $\bar{B}$ are also independent.''
We model the waiting time, expressed in minutes, at a counter, by a random variable $T$ that follows the exponential distribution with parameter 0.7. Statement $\mathbf{n^o 3}$: ``The probability that a customer waits at least five minutes at this counter is approximately 0.7.'' Statement $\mathbf{n^o 4}$: ``The average waiting time at this counter is seven minutes.''
We know that $39\%$ of the French population has blood group A+. We want to know if this proportion is the same among blood donors. We survey 183 blood donors and among them, $34\%$ have blood group A+. Statement $\mathbf{n^o 5}$: ``We cannot reject, at the $5\%$ significance level, the hypothesis that the proportion of people with blood group A+ among blood donors is $39\%$ as in the general population.''
For each of the following five statements, indicate whether it is true or false and justify the answer.\\
An unjustified answer is not taken into account. An absence of answer is not penalized.
\begin{enumerate}
\item Zoé goes to work on foot or by car. Where she lives, it rains one day out of four.\\
When it rains, Zoé goes to work by car in $80\%$ of cases.\\
When it does not rain, she goes to work on foot with a probability equal to 0.6.
\textbf{Statement $\mathbf{n^o 1}$:}\\
``Zoé uses the car one day out of two.''\\
\item In the set $E$ of outcomes of a random experiment, we consider two events $A$ and $B$.
\textbf{Statement $\mathbf{n^o 2}$:}\\
``If $A$ and $B$ are independent, then $A$ and $\bar{B}$ are also independent.''\\
\item We model the waiting time, expressed in minutes, at a counter, by a random variable $T$ that follows the exponential distribution with parameter 0.7.
\textbf{Statement $\mathbf{n^o 3}$:}\\
``The probability that a customer waits at least five minutes at this counter is approximately 0.7.''
\textbf{Statement $\mathbf{n^o 4}$:}\\
``The average waiting time at this counter is seven minutes.''\\
\item We know that $39\%$ of the French population has blood group A+.\\
We want to know if this proportion is the same among blood donors.\\
We survey 183 blood donors and among them, $34\%$ have blood group A+.
\textbf{Statement $\mathbf{n^o 5}$:}\\
``We cannot reject, at the $5\%$ significance level, the hypothesis that the proportion of people with blood group A+ among blood donors is $39\%$ as in the general population.''
\end{enumerate}