In this exercise, we are interested in the operating mode of two restaurants: without reservation or with prior reservation.
The first restaurant operates without reservation but the waiting time to obtain a table is often a problem for customers. We model this waiting time in minutes by a random variable $X$ which follows an exponential distribution with parameter $\lambda$ where $\lambda$ is a strictly positive real number. We recall that the mathematical expectation of $X$ is equal to $\frac{1}{\lambda}$. A statistical study made it possible to observe that the average waiting time to obtain a table is 10 minutes. a. Determine the value of $\lambda$. b. What is the probability that a customer waits between 10 and 20 minutes to obtain a table? Round to $10^{-4}$. c. A customer has been waiting for 10 minutes. What is the probability that he must wait at least 5 more minutes to obtain a table? Round to $10^{-4}$.
The second restaurant has a capacity of 70 seats and only serves people who have made a prior reservation. The probability that a person who has made a reservation appears at the restaurant is estimated at 0.8. We denote by $n$ the number of reservations taken by the restaurant and $Y$ the random variable corresponding to the number of people who have made a reservation and appear at the restaurant. We admit that the behaviours of people who have made a reservation are independent of each other. The random variable $Y$ then follows a binomial distribution. a. Specify, as a function of $n$, the parameters of the distribution of the random variable $Y$, its mathematical expectation $E(Y)$ and its standard deviation $\sigma(Y)$. b. In this question, we denote by $Z$ a random variable following the normal distribution $\mathscr{N}\left(\mu, \sigma^{2}\right)$ with mean $\mu = 64.8$ and standard deviation $\sigma = 3.6$. Calculate the probability $p_{1}$ of the event $\{Z \leqslant 71\}$ using a calculator. c. We admit that when $n = 81$, $p_{1}$ is an approximate value to within $10^{-2}$ of the probability $p(Y \leqslant 70)$ of the event $\{Z \leqslant 70\}$. The restaurant has received 81 reservations. What is the probability that it cannot accommodate some of the customers who have made a reservation and appear?
In this exercise, we are interested in the operating mode of two restaurants: without reservation or with prior reservation.
\begin{enumerate}
\item The first restaurant operates without reservation but the waiting time to obtain a table is often a problem for customers.\\
We model this waiting time in minutes by a random variable $X$ which follows an exponential distribution with parameter $\lambda$ where $\lambda$ is a strictly positive real number. We recall that the mathematical expectation of $X$ is equal to $\frac{1}{\lambda}$.\\
A statistical study made it possible to observe that the average waiting time to obtain a table is 10 minutes.\\
a. Determine the value of $\lambda$.\\
b. What is the probability that a customer waits between 10 and 20 minutes to obtain a table? Round to $10^{-4}$.\\
c. A customer has been waiting for 10 minutes. What is the probability that he must wait at least 5 more minutes to obtain a table? Round to $10^{-4}$.
\item The second restaurant has a capacity of 70 seats and only serves people who have made a prior reservation. The probability that a person who has made a reservation appears at the restaurant is estimated at 0.8.\\
We denote by $n$ the number of reservations taken by the restaurant and $Y$ the random variable corresponding to the number of people who have made a reservation and appear at the restaurant.\\
We admit that the behaviours of people who have made a reservation are independent of each other. The random variable $Y$ then follows a binomial distribution.\\
a. Specify, as a function of $n$, the parameters of the distribution of the random variable $Y$, its mathematical expectation $E(Y)$ and its standard deviation $\sigma(Y)$.\\
b. In this question, we denote by $Z$ a random variable following the normal distribution $\mathscr{N}\left(\mu, \sigma^{2}\right)$ with mean $\mu = 64.8$ and standard deviation $\sigma = 3.6$.\\
Calculate the probability $p_{1}$ of the event $\{Z \leqslant 71\}$ using a calculator.\\
c. We admit that when $n = 81$, $p_{1}$ is an approximate value to within $10^{-2}$ of the probability $p(Y \leqslant 70)$ of the event $\{Z \leqslant 70\}$.\\
The restaurant has received 81 reservations.\\
What is the probability that it cannot accommodate some of the customers who have made a reservation and appear?
\end{enumerate}