An industrial bakery uses a machine to manufacture loaves of country bread weighing on average 400 grams. To be sold to customers, these loaves must weigh at least 385 grams. A loaf whose mass is strictly less than 385 grams is non-marketable, a loaf whose mass is greater than or equal to 385 grams is marketable. The mass of a loaf manufactured by the machine can be modeled by a random variable $X$ following the normal distribution with mean $\mu = 400$ and standard deviation $\sigma = 11$.
Probabilities will be rounded to the nearest thousandth.

Part A

You may use the following table in which values are rounded to the nearest thousandth.

$x$	380	385	390	395	400	405	410	415	420
$P ( X \leqslant x )$	0,035	0,086	0,182	0,325	0,5	0,675	0,818	0,914	0,965

1. Calculate $P ( 390 \leqslant X \leqslant 410 )$.
2. Calculate the probability $p$ that a loaf chosen at random from production is marketable.
3. The manufacturer finds this probability $p$ too low. He decides to modify his production methods in order to vary the value of $\sigma$ without changing that of $\mu$. For what value of $\sigma$ is the probability that a loaf is marketable equal to $96\%$ ? Round the result to the nearest tenth. You may use the following result: when $Z$ is a random variable that follows the normal distribution with mean 0 and standard deviation 1, we have $P ( Z \leqslant - 1,751 ) \approx 0,040$.

Part B

The production methods have been modified with the aim of obtaining $96\%$ marketable loaves. To evaluate the effectiveness of these modifications, a quality control is performed on a sample of 300 loaves manufactured.

1. Determine the asymptotic confidence interval at the $95\%$ confidence level for the proportion of marketable loaves in a sample of size 300.
2. Among the 300 loaves in the sample, 283 are marketable.

In light of the confidence interval obtained in question 1, can we decide that the objective has been achieved?

Part C

The baker uses an electronic scale. The operating time without malfunction, in days, of this electronic scale is a random variable $T$ that follows an exponential distribution with parameter $\lambda$.

1. We know that the probability that the electronic scale does not malfunction before 30 days is 0,913. Deduce the value of $\lambda$ rounded to the nearest thousandth.

Throughout the rest, we will take $\lambda = 0,003$.
2. What is the probability that the electronic scale continues to function without malfunction after 90 days, given that it has functioned without malfunction for 60 days?
3. The seller of this electronic scale assured the baker that there was a one in two chance that the scale would not malfunction before a year. Is he right? If not, for how many days is this true?

In this part, the calculated probabilities will be rounded to the nearest thousandth. The industrialist markets his valves to many customers. Monthly demand is a random variable $D$ that follows the normal distribution with mean $\mu = 800$ and standard deviation $\sigma = 40$.

1. Determine $P(760 \leqslant D \leqslant 840)$.
2. Determine $P(D \leqslant 880)$.
3. The industrialist thinks that if he builds a monthly stock of 880 valves, he will have no more than a $1\%$ chance of running out of stock. Is he right?

According to a statistical study conducted over several months, it is assumed that the number $X$ of budget padlocks sold per month in the hardware store can be modeled by a random variable that follows a normal distribution with mean $\mu = 750$ and standard deviation $\sigma = 25$.
Calculate $P(725 \leqslant X \leqslant 775)$.

Let $X$ be a random variable that follows the normal distribution with mean $\mu = 110$ and standard deviation $\sigma = 25$. What is the value rounded to the nearest thousandth of the probability $P ( X \geqslant 135 )$ ? a. 0.159 b. 0.317 c. 0.683 d. 0.841

In an industrial bakery, a baguette is randomly selected from production. It is admitted that the random variable expressing its mass, in grams, follows the normal distribution with mean 200 and standard deviation 10.
Statement 1: The probability that the mass of the baguette is greater than 187 g is greater than 0.9.
Indicate whether this statement is true or false, justifying your answer.

Exercise 3 (3 points)

Part A:

A health control agency is interested in the number of bacteria of a certain type contained in fresh cream. It performs analyses on 10000 samples of 1 ml of fresh cream from the entire French production. The results are given in the table below:

\begin{tabular}{ l } Number of bacteria

(in thousands)

& $[100;120[$ & $[120;130[$ & $[130;140[$ & $[140;150[$ & $[150;160[$ & $[160;180[$ \hline Number of samples & 1597 & 1284 & 2255 & 1808 & 1345 & 1711 \hline \end{tabular}
Using a calculator, give an estimate of the mean and standard deviation of the number of bacteria per sample.

Part B:

The agency then decides to model the number of bacteria studied (in thousands per ml) present in fresh cream by a random variable $X$ following the normal distribution with parameters $\mu = 140$ and $\sigma = 19$.

1. a. Is this choice of modelling relevant? Argue. b. We denote $p = P(X \geqslant 160)$. Determine the value of $p$ rounded to $10^{-3}$.
   
2. During the inspection of a dairy, the health control agency analyzes a sample of 50 samples of 1 ml of fresh cream from the production of this dairy; 13 samples contain more than 160 thousand bacteria. a. The agency declares that there is an anomaly in the production and that it can affirm it with a probability of 0.05 of being wrong. Justify its declaration. b. Could it have affirmed it with a probability of 0.01 of being wrong?

Exercise 1 (6 points)

We study certain characteristics of a supermarket in a small town.

Part A - Preliminary Demonstration

Let $X$ be a random variable that follows the exponential distribution with parameter 0.2. Recall that the expectation of the random variable $X$, denoted $E(X)$, is equal to: $$\lim_{x \rightarrow +\infty} \int_{0}^{x} 0.2t\, \mathrm{e}^{-0.2t} \mathrm{~d}t$$ The purpose of this part is to demonstrate that $E(X) = 5$.

1. Let $g$ be the function defined on the interval $[0; +\infty[$ by $g(t) = 0.2t\,\mathrm{e}^{-0.2t}$.
   We define the function $G$ on the interval $[0; +\infty[$ by $G(t) = (-t-5)\mathrm{e}^{-0.2t}$. Verify that $G$ is a primitive of $g$ on the interval $[0; +\infty[$.
2. Deduce that the exact value of $E(X)$ is 5.
   Hint: you may use, without proving it, the following result: $$\lim_{x \rightarrow +\infty} x\,\mathrm{e}^{-0.2x} = 0$$

Part B - Study of the duration of a customer's presence in the supermarket

A study commissioned by the supermarket manager makes it possible to model the duration, expressed in minutes, spent in the supermarket by a randomly chosen customer using a random variable $T$. This variable $T$ follows a normal distribution with expectation 40 minutes and standard deviation a positive real number denoted $\sigma$. Thanks to this study, it is estimated that $P(T < 10) = 0.067$.

1. Determine an approximate value of the real number $\sigma$ to the nearest second.
2. In this question, we take $\sigma = 20$ minutes. What is then the proportion of customers who spend more than one hour in the supermarket?

Part C - Waiting time for payment

This supermarket gives customers the choice to use automatic payment terminals alone or to go through a checkout managed by an operator.

1. The waiting time at an automatic terminal, expressed in minutes, is modeled by a random variable that follows the exponential distribution with parameter $0.2\,\mathrm{min}^{-1}$. a. Give the average waiting time for a customer at an automatic payment terminal. b. Calculate the probability, rounded to $10^{-3}$, that the waiting time for a customer at an automatic payment terminal is greater than 10 minutes.
2. The study commissioned by the manager leads to the following modeling:
   • among customers who chose to use an automatic terminal, 86\% wait less than 10 minutes;
   • among customers using a checkout, 63\% wait less than 10 minutes.
   We randomly choose a customer from the store and define the following events: $B$: ``the customer pays at an automatic terminal''; $\bar{B}$: ``the customer pays at a checkout with an operator''; $S$: ``the customer's waiting time during payment is less than 10 minutes''.
   A waiting time greater than ten minutes at a checkout with an operator or at an automatic terminal creates a negative perception of the store in the customer. The manager wants more than 75\% of customers to wait less than 10 minutes. What is the minimum proportion of customers who must choose an automatic payment terminal for this objective to be achieved?

Part D - Gift vouchers

During payment, scratch cards, winning or losing, are distributed to customers. The number of cards distributed depends on the amount of purchases. Each customer receives one scratch card per 10~\euro{} of purchases. For example, if the purchase amount is 58.64~\euro{}, then the customer receives 5 cards; if the amount is 124.31~\euro{}, the customer receives 12 cards. Winning cards represent 0.5\% of the entire stock of cards. Furthermore, this stock is large enough to treat the distribution of a card as a draw with replacement.

1. A customer makes purchases for an amount of 158.02~\euro{}.
   What is the probability, rounded to $10^{-2}$, that they obtain at least one winning card?
2. From what purchase amount, rounded to 10~\euro{}, is the probability of obtaining at least one winning card greater than 50\%?

The number of trees per hectare in this forest can be modelled by a random variable $X$ following a normal distribution with mean $\mu = 4000$ and standard deviation $\sigma = 300$.

1. Determine the probability that there are between 3400 and 4600 trees on a given hectare of this forest. The result will be given rounded to $10^{-3}$.
2. Calculate the probability that there are more than 4500 trees on a given hectare of this forest. The result will be given rounded to $10^{-3}$.

A company packages white sugar from two farms $U$ and $V$ in 1 kg packets of different qualities. Extra fine sugar is packaged separately in packets bearing the label ``extra fine''. Throughout the exercise, results should be rounded, if necessary, to the nearest thousandth.
To calibrate the sugar according to the size of its crystals, it is passed through a series of three sieves. Sugar crystals with a size less than $0.2\,\mathrm{mm}$ are found in the sealed bottom container at the end of calibration. They will be packaged in packets bearing the label ``extra fine sugar''.

1. A sugar crystal is randomly selected from farm U. The size of this crystal, expressed in millimeters, is modeled by the random variable $X_{\mathrm{U}}$ which follows the normal distribution with mean $\mu_{\mathrm{U}} = 0.58\,\mathrm{mm}$ and standard deviation $\sigma_{\mathrm{U}} = 0.21\,\mathrm{mm}$. a. Calculate the probabilities of the following events: $X_{\mathrm{U}} < 0.2$ and $0.5 \leqslant X_{\mathrm{U}} < 0.8$. b. 1800 grams of sugar from farm $U$ is passed through the series of sieves. Deduce from the previous question an estimate of the mass of sugar recovered in the sealed bottom container and an estimate of the mass of sugar recovered in sieve 2.
2. A sugar crystal is randomly selected from farm V. The size of this crystal, expressed in millimeters, is modeled by the random variable $X_{\mathrm{V}}$ which follows the normal distribution with mean $\mu_{\mathrm{V}} = 0.65\,\mathrm{mm}$ and standard deviation $\sigma_{\mathrm{V}}$ to be determined. During the calibration of a large quantity of sugar crystals from farm V, it is observed that $40\%$ of these crystals end up in sieve 2. What is the value of the standard deviation $\sigma_{\mathrm{V}}$ of the random variable $X_{\mathrm{V}}$?

A factory manufactures tubes.
Part A
Questions 1. and 2. are independent. We are interested in two types of tubes, called type 1 tubes and type 2 tubes.

1. A type 1 tube is accepted at inspection if its thickness is between 1.35 millimetres and 1.65 millimetres. a. Let $X$ denote the random variable that, for each type 1 tube randomly selected from the day's production, gives its thickness expressed in millimetres. We assume that the random variable $X$ follows a normal distribution with mean 1.5 and standard deviation 0.07.
   A type 1 tube is randomly selected from the day's production. Calculate the probability that the tube is accepted at inspection. b. The company wishes to improve the quality of type 1 tube production. To do this, the settings of the machines producing these tubes are modified. Let $X _ { 1 }$ denote the random variable that, for each type 1 tube selected from the production of the modified machine, gives its thickness. We assume that the random variable $X _ { 1 }$ follows a normal distribution with mean 1.5 and standard deviation $\sigma _ { 1 }$.
   A type 1 tube is randomly selected from the production of the modified machine. Determine an approximate value to $10 ^ { - 3 }$ of $\sigma _ { 1 }$ so that the probability that this tube is accepted at inspection equals 0.98. (You may use the random variable $Z$ defined by $Z = \frac { X _ { 1 } - 1.5 } { \sigma _ { 1 } }$ which follows the standard normal distribution.)
   
2. A machine produces type 2 tubes. A type 2 tube is said to be ``compliant for length'' when its length, in millimetres, belongs to the interval [298; 302]. The specifications establish that, in the production of type 2 tubes, a proportion of $2 \%$ of tubes that are not ``compliant for length'' is acceptable.
   It is desired to decide whether the production machine should be serviced. To do this, a random sample of 250 tubes is taken from the production of type 2 tubes, in which 10 tubes are found to be not ``compliant for length''. a. Give an asymptotic confidence interval at $95 \%$ for the frequency of tubes not ``compliant for length'' in a sample of 250 tubes. b. Should the machine be serviced? Justify your answer.

Part B
Adjustment errors in the production line can affect the thickness or length of type 2 tubes.
A study conducted on the production revealed that:

• $96 \%$ of type 2 tubes have compliant thickness;
• among type 2 tubes that have compliant thickness, $95 \%$ have compliant length;
• $3.6 \%$ of type 2 tubes have non-compliant thickness and compliant length.

A type 2 tube is randomly selected from the production and we consider the events: --- $E$: ``the thickness of the tube is compliant''; --- $L$: ``the length of the tube is compliant''. We model the random experiment with a probability tree.

1. Copy and complete this tree entirely.
2. Show that the probability of event $L$ equals 0.948.

A company specializes in the sale of tiles.
Parts A, B and C are independent.

Part A

We assume in this part that the company sells batches of tiles containing $25\%$ of tiles with pattern and $75\%$ of white tiles. During a quality control, it is observed that:

• $2.25\%$ of the tiles are cracked;
• $6\%$ of the tiles with pattern are cracked.

A tile is randomly selected. We denote by $M$ the event ``the tile has a pattern'' and $F$ the event ``the tile is cracked''.

1. Translate the situation using a probability tree.
2. We know that the selected tile is cracked. Prove that the probability that it is a tile with pattern is $\frac{2}{3}$.
3. Calculate $P_{\bar{M}}(F)$, the probability of $F$ given $\bar{M}$.

Part B

We model the thickness in millimeters of a randomly selected tile by a random variable $X$ that follows a normal distribution with mean $\mu = 11$ and standard deviation $\sigma$.
A tile is marketable if its thickness measures between $10.1\text{ mm}$ and $11.9\text{ mm}$. We know that $99\%$ of the tiles are marketable.

1. Prove that $P(X < 10.1) = 0.005$.
2. We introduce the random variable $Z$ such that $$Z = \frac{X - 11}{\sigma}.$$ a. Give the distribution followed by the random variable $Z$. b. Prove that $P\left(Z \leqslant -\frac{0.9}{\sigma}\right) = 0.005$. c. Deduce the value of $\sigma$ rounded to the nearest hundredth.

Part C

We consider the function $f$ defined on $[0; 2\pi]$ by $$f(x) = -1.5\cos(x) + 1.5$$ We admit that the function $f$ is continuous on $[0; 2\pi]$. We denote by $\mathscr{C}_1$ the representative curve of the function $f$ in an orthonormal coordinate system.

1. Prove that the function $f$ is positive on $[0; 2\pi]$.
2. In the figure above, the curve drawn in dashes, denoted $\mathscr{C}_2$, is the curve symmetric to $\mathscr{C}_1$ with respect to the $x$-axis. The shape of a tile is that of the region bounded by the curves $\mathscr{C}_1$ and $\mathscr{C}_2$. We denote by $\mathscr{A}$ its area, expressed in square units. Calculate $\mathscr{A}$.

In this exercise and unless otherwise stated, results should be rounded to $10^{-3}$.
A factory manufactures tubes.

Part A

Questions 1. and 2. are independent. We are interested in two types of tubes, called type 1 tubes and type 2 tubes.

1. A type 1 tube is accepted at inspection if its thickness is between 1.35 millimetres and 1.65 millimetres. a. Let $X$ denote the random variable which, for each type 1 tube randomly selected from the day's production, gives its thickness expressed in millimetres. We assume that the random variable $X$ follows a normal distribution with mean 1.5 and standard deviation 0.07.
   A type 1 tube is randomly selected from the day's production. Calculate the probability that the tube is accepted at inspection. b. The company wishes to improve the quality of type 1 tube production. To do this, the settings of the machines producing these tubes are modified. Let $X_1$ denote the random variable which, for each type 1 tube selected from the production of the modified machine, gives its thickness. We assume that the random variable $X_1$ follows a normal distribution with mean 1.5 and standard deviation $\sigma_1$.
   A type 1 tube is randomly selected from the production of the modified machine. Determine an approximate value to $10^{-3}$ of $\sigma_1$ so that the probability that this tube is accepted at inspection equals 0.98. (You may use the random variable $Z$ defined by $Z = \frac{X_1 - 1.5}{\sigma_1}$ which follows the standard normal distribution.)
   
2. A machine produces type 2 tubes. A type 2 tube is said to be ``compliant for length'' when its length, in millimetres, belongs to the interval [298; 302]. The specifications establish that, in the production of type 2 tubes, a proportion of $2\%$ of tubes that are not ``compliant for length'' is acceptable.
   It is desired to decide whether the production machine should be serviced. To do this, a random sample of 250 tubes is taken from the production of type 2 tubes, in which 10 tubes are found to be not ``compliant for length''. a. Give an asymptotic confidence interval at $95\%$ for the frequency of tubes not ``compliant for length'' in a sample of 250 tubes. b. Should the machine be serviced? Justify your answer.

Part B

Adjustment errors in the production line can affect the thickness or length of type 2 tubes.
A study conducted on the production revealed that: --- $96\%$ of type 2 tubes have compliant thickness; --- among type 2 tubes that have compliant thickness, $95\%$ have compliant length; --- $3.6\%$ of type 2 tubes have non-compliant thickness and compliant length.
A type 2 tube is randomly selected from the production and we consider the events: --- $E$: ``the tube's thickness is compliant''; --- $L$: ``the tube's length is compliant''.
We model the random experiment with a probability tree.

1. Copy and complete this tree entirely.
2. Show that the probability of event $L$ equals 0.948.

At a certain car wash, the time required to wash one car follows a normal distribution with mean 30 minutes and standard deviation 2 minutes. When washing one car at this car wash, what is the probability that the washing time is 33 minutes or more, using the standard normal distribution table below? [3 points]

$z$	$\mathrm { P } ( 0 \leqq Z \leqq z )$
0.5	0.1915
1.0	0.3413
1.5	0.4332
2.0	0.4772

(1) 0.0228
(2) 0.0668
(3) 0.1587
(4) 0.2708
(5) 0.3085

The test scores of all students at a certain school follow a normal distribution with mean 500 and standard deviation 25. When one student is randomly selected from this school, what is the probability that the student's test score is at least 475 and at most 550, using the standard normal distribution table below? [3 points]

$z$	$\mathrm{P}(0 \leq Z \leq z)$
1.0	0.3413
1.5	0.4332
2.0	0.4772
2.5	0.4938

(1) 0.7745
(2) 0.8185
(3) 0.9104
(4) 0.9270
(5) 0.9710

A research institute investigated the length of tomato seedling stems 3 weeks after planting. The length of the tomato stems follows a normal distribution with mean 30 cm and standard deviation 2 cm. Using the standard normal distribution table on the right, find the probability that the length of a randomly selected tomato stem is at least 27 cm and at most 32 cm. [3 points]

$z$	$\mathrm { P } ( 0 \leq Z \leq z )$
1.0	0.3413
1.5	0.4332
2.0	0.4772
2.5	0.4938

(1) 0.6826
(2) 0.7745
(3) 0.8185
(4) 0.9104
(5) 0.9270

A snack factory produces snacks where the weight of one package follows a normal distribution with mean 75 g and standard deviation 2 g. Using the standard normal distribution table below, what is the probability that the weight of a randomly selected package of snacks from this factory is at least 76 g and at most 78 g? [3 points]

$z$	$\mathrm { P } ( 0 \leq Z \leq z )$
0.5	0.1915
1.0	0.3413
1.5	0.4332
2.0	0.4772

(1) 0.0440
(2) 0.0919
(3) 0.1359
(4) 0.1498
(5) 0.2417

At a rice collection event, the weight of rice donated by each student follows a normal distribution with mean 1.5 kg and standard deviation 0.2 kg. When one student is randomly selected from those who participated in the event, what is the probability that the weight of rice donated by this student is at least 1.3 kg and at most 1.8 kg, using the standard normal distribution table below? [3 points]

$z$	$\mathrm { P } ( 0 \leq Z \leq z )$
1.00	0.3413
1.25	0.3944
1.50	0.4332
1.75	0.4599

(1) 0.8543
(2) 0.8012
(3) 0.7745
(4) 0.7357
(5) 0.6826

The weight of one paprika harvested at a certain farm follows a normal distribution with mean 180 g and standard deviation 20 g. Using the standard normal distribution table below, what is the probability that the weight of one randomly selected paprika from this farm is at least 190 g and at most 210 g? [3 points]

$z$	$\mathrm { P } ( 0 \leq Z \leq z )$
0.5	0.1915
1.0	0.3413
1.5	0.4332
2.0	0.4772

(1) 0.0440
(2) 0.0919
(3) 0.1359
(4) 0.1498
(5) 0.2417