Part A: Study of the lifespan of a household appliance
Statistical studies have made it possible to model the lifespan, in months, of a type of dishwasher by a random variable $X$ following a normal distribution $\mathscr{N}(\mu, \sigma^2)$ with mean $\mu = 84$ and standard deviation $\sigma$. Furthermore, we have $P(X \leqslant 64) = 0.16$.
a. By exploiting the graph, determine $P(64 \leqslant X \leqslant 104)$. b. What approximate integer value of $\sigma$ can we propose?
We denote by $Z$ the random variable defined by $Z = \dfrac{X - 84}{\sigma}$. a. What is the probability distribution followed by $Z$? b. Justify that $P(X \leqslant 64) = P\!\left(Z \leqslant \dfrac{-20}{\sigma}\right)$. c. Deduce the value of $\sigma$, rounded to $10^{-3}$.
In this question, we consider that $\sigma = 20.1$. The probabilities requested will be rounded to $10^{-3}$. a. Calculate the probability that the lifespan of the dishwasher is between 2 and 5 years. b. Calculate the probability that the dishwasher has a lifespan greater than 10 years.
Part B: Study of the warranty extension offered by El'Ectro
The dishwasher is guaranteed free of charge for the first two years. The company El'Ectro offers its customers a warranty extension of 3 additional years. Statistical studies conducted on customers who take the warranty extension show that $11.5\%$ of them use the warranty extension.
We randomly choose 12 customers among those who have taken the warranty extension (this choice can be treated as random sampling with replacement given the large number of customers). a. What is the probability that exactly 3 of these customers use this warranty extension? Detail the approach by specifying the probability distribution used. Round to $10^{-3}$. b. What is the probability that at least 6 of these customers use this warranty extension? Round to $10^{-3}$.
The warranty extension offer is as follows: for 65 euros additional, El'Ectro will reimburse the customer the initial value of the dishwasher, namely 399 euros, if an irreparable breakdown occurs between the beginning of the third year and the end of the fifth year. The customer cannot use this warranty extension if the breakdown is repairable. We randomly choose a customer among those who have subscribed to the warranty extension, and we denote by $Y$ the random variable representing the algebraic gain in euros realized on this customer by the company El'Ectro, thanks to the warranty extension. a. Justify that $Y$ takes the values 65 and $-334$ then give the probability distribution of $Y$. b. Is this warranty extension offer financially advantageous for the company? Justify.
\section*{Exercise 3 -- Common to all candidates}
\section*{Part A: Study of the lifespan of a household appliance}
Statistical studies have made it possible to model the lifespan, in months, of a type of dishwasher by a random variable $X$ following a normal distribution $\mathscr{N}(\mu, \sigma^2)$ with mean $\mu = 84$ and standard deviation $\sigma$. Furthermore, we have $P(X \leqslant 64) = 0.16$.
\begin{enumerate}
\item a. By exploiting the graph, determine $P(64 \leqslant X \leqslant 104)$.\\
b. What approximate integer value of $\sigma$ can we propose?
\item We denote by $Z$ the random variable defined by $Z = \dfrac{X - 84}{\sigma}$.\\
a. What is the probability distribution followed by $Z$?\\
b. Justify that $P(X \leqslant 64) = P\!\left(Z \leqslant \dfrac{-20}{\sigma}\right)$.\\
c. Deduce the value of $\sigma$, rounded to $10^{-3}$.
\item In this question, we consider that $\sigma = 20.1$.
The probabilities requested will be rounded to $10^{-3}$.\\
a. Calculate the probability that the lifespan of the dishwasher is between 2 and 5 years.\\
b. Calculate the probability that the dishwasher has a lifespan greater than 10 years.
\end{enumerate}
\section*{Part B: Study of the warranty extension offered by El'Ectro}
The dishwasher is guaranteed free of charge for the first two years. The company El'Ectro offers its customers a warranty extension of 3 additional years. Statistical studies conducted on customers who take the warranty extension show that $11.5\%$ of them use the warranty extension.
\begin{enumerate}
\item We randomly choose 12 customers among those who have taken the warranty extension (this choice can be treated as random sampling with replacement given the large number of customers).\\
a. What is the probability that exactly 3 of these customers use this warranty extension? Detail the approach by specifying the probability distribution used. Round to $10^{-3}$.\\
b. What is the probability that at least 6 of these customers use this warranty extension? Round to $10^{-3}$.
\item The warranty extension offer is as follows: for 65 euros additional, El'Ectro will reimburse the customer the initial value of the dishwasher, namely 399 euros, if an irreparable breakdown occurs between the beginning of the third year and the end of the fifth year. The customer cannot use this warranty extension if the breakdown is repairable.
We randomly choose a customer among those who have subscribed to the warranty extension, and we denote by $Y$ the random variable representing the algebraic gain in euros realized on this customer by the company El'Ectro, thanks to the warranty extension.\\
a. Justify that $Y$ takes the values 65 and $-334$ then give the probability distribution of $Y$.\\
b. Is this warranty extension offer financially advantageous for the company? Justify.
\end{enumerate}