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?

2. Let $Y$ denote the random variable which, for a small jar randomly selected from the production of line $\mathrm { F } _ { 2 }$, associates its sugar content. We assume that $Y$ follows the normal distribution with mean $m _ { 2 } = 0.17$ and standard deviation $\sigma _ { 2 }$. We further assume that the probability that a small jar randomly selected from the production of line $\mathrm { F } _ { 2 }$ is compliant equals 0.99 . Let Z be the random variable defined by $Z = \frac { Y - m _ { 2 } } { \sigma _ { 2 } }$. a. What distribution does the random variable $Z$ follow? b. Determine, as a function of $\sigma _ { 2 }$, the interval to which $Z$ belongs when $Y$ belongs to the interval $[ 0.16 ; 0.18 ]$. c. Deduce an approximate value to $10 ^ { - 3 }$ near of $\sigma _ { 2 }$.
You may use the table given below, in which the random variable $Z$ follows the normal distribution with mean 0 and standard deviation 1 .

$\beta$	$P ( - \beta \leqslant Z \leqslant \beta )$
2.4324	0.985
2.4573	0.986
2.4838	0.987
2.5121	0.988
2.5427	0.989
2.5758	0.990
2.6121	0.991
2.6521	0.992
2.6968	0.993

Exercise 3

Common to all candidates

Given a real number $k$, we consider the function $f _ { k }$ defined on $\mathbb { R }$ by
$$f _ { k } ( x ) = \frac { 1 } { 1 + \mathrm { e } ^ { - k x } }$$
The plane is equipped with an orthonormal coordinate system ( $\mathrm { O } , \vec { \imath } , \vec { \jmath }$ ).

Part A

In this part we choose $k = 1$. We have therefore, for all real $x , f _ { 1 } ( x ) = \frac { 1 } { 1 + \mathrm { e } ^ { - x } }$. The graph $\mathscr { C } _ { 1 }$ of the function $f _ { 1 }$ in the coordinate system ( $\mathrm { O } , \vec { \imath } , \vec { \jmath }$ ) is given in APPENDIX, to be returned with your answer sheet.

1. Determine the limits of $f _ { 1 } ( x )$ as $x \to + \infty$ and as $x \to - \infty$ and give a graphical interpretation of the results obtained.
2. Prove that, for all real $x , f _ { 1 } ( x ) = \frac { \mathrm { e } ^ { x } } { 1 + \mathrm { e } ^ { x } }$.
3. Let $f _ { 1 } ^ { \prime }$ denote the derivative function of $f _ { 1 }$ on $\mathbb { R }$. Calculate, for all real $x , f _ { 1 } ^ { \prime } ( x )$.

Deduce the variations of the function $f _ { 1 }$ on $\mathbb { R }$.

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?

A factory manufactures spherical balls whose diameter is expressed in millimetres. A ball is said to be out of specification when its diameter is less than 9 mm or greater than 11 mm.
Part A
1. Let $X$ be the random variable that associates to each ball chosen at random from production its diameter expressed in mm.
It is assumed that the random variable $X$ follows the normal distribution with mean 10 and standard deviation 0.4.
Show that an approximate value to 0.0001 of the probability that a ball is out of specification is 0.0124. You may use the table of values given in the appendix.
2. A production control is put in place such that 98\% of out-of-specification balls are rejected and 99\% of correct balls are kept.
A ball is chosen at random from production. Let $N$ denote the event: ``the chosen ball is within specification'', and $A$ the event: ``the chosen ball is accepted after the control''.
a. Construct a weighted tree diagram that incorporates the data from the problem statement.
b. Calculate the probability of event $A$.
c. What is the probability that an accepted ball is out of specification?
Part B
This production control proving too costly for the company, it is abandoned: henceforth, all balls produced are kept, and they are packaged in bags of 100 balls.
It is considered that the probability that a ball is out of specification is 0.0124.
It will be assumed that taking a bag of 100 balls at random is equivalent to performing a sampling with replacement of 100 balls from the set of manufactured balls.
Let $Y$ be the random variable that associates to every bag of 100 balls the number of out-of-specification balls in that bag.
1. What is the distribution followed by the random variable $Y$?
2. What are the mean and standard deviation of the random variable $Y$?
3. What is the probability that a bag of 100 balls contains exactly two out-of-specification balls?
4. What is the probability that a bag of 100 balls contains at most one out-of-specification ball?

Exercise 1 (5 points)

A large cosmetics brand launches a new moisturizing cream.

Part A: Packaging of jars

This brand wishes to sell the new cream in a 50 mL package and has jars with a maximum capacity of 55 mL for this purpose.
A jar of cream is said to be non-compliant if it contains less than 49 mL of cream.

1. Several series of tests lead to modeling the quantity of cream, expressed in mL, contained in each jar by a random variable $X$ which follows the normal distribution with mean $\mu = 50$ and standard deviation $\sigma = 1.2$. Calculate the probability that a jar of cream is non-compliant.
2. The proportion of non-compliant jars of cream is judged to be too large. By modifying the viscosity of the cream, we can change the value of the standard deviation of the random variable $X$, without modifying its mean $\mu = 50$. We want to reduce to 0.06 the probability that a randomly chosen jar is non-compliant. We denote $\sigma ^ { \prime }$ the new standard deviation, and $Z$ the random variable equal to $\frac { X - 50 } { \sigma ^ { \prime } }$ a. Specify the distribution followed by the random variable $Z$. b. Determine an approximate value of the real number $u$ such that $p ( Z \leqslant u ) = 0.06$. c. Deduce the expected value of $\sigma ^ { \prime }$.
3. A shop orders 50 jars of this new cream from its supplier.

We consider that the work on the viscosity of the cream has made it possible to achieve the set objective and therefore that the proportion of non-compliant jars in the sample is 0.06. Let $Y$ be the random variable equal to the number of non-compliant jars among the 50 jars received. a. We admit that $Y$ follows a binomial distribution. Give its parameters. b. Calculate the probability that the shop receives two non-compliant jars or fewer than two non-compliant jars.

Part B: Advertising campaign

A consumer association decides to estimate the proportion of people satisfied by the use of this cream. It conducts a survey among people using this product. Out of 140 people interviewed, 99 declare themselves satisfied. Estimate, by confidence interval at the 95\% threshold, the proportion of satisfied people among the users of the cream.

A football is compliant with regulations if it meets, depending on its size, two conditions simultaneously (on its mass and on its circumference). In particular, a standard-sized football is compliant with regulations when its mass, expressed in grams, belongs to the interval [410;450] and its circumference, expressed in centimetres, belongs to the interval [68;70].

1. Let $X$ denote the random variable which, for each standard-sized football chosen at random in the company, associates its mass in grams. It is admitted that $X$ follows a normal distribution with mean 430 and standard deviation 10. Determine an approximate value to $10 ^ { - 3 }$ of the probability $P ( 410 \leqslant X \leqslant 450 )$.
2. Let $Y$ denote the random variable which, for each standard-sized football chosen at random in the company, associates its circumference in centimetres. It is admitted that $Y$ follows a normal distribution with mean 69 and standard deviation $\sigma$. Determine the value of $\sigma$, to the nearest hundredth, knowing that $97 \%$ of standard-sized footballs have a circumference compliant with regulations. You may use the following result: when $Z$ is a random variable that follows the standard normal distribution, then $P ( - \beta \leqslant Z \leqslant \beta ) = 0,97$ for $\beta \approx 2,17$.

Each cone is filled with vanilla ice cream. We denote by $Y$ the random variable which, to each cone, associates the mass (expressed in grams) of ice cream it contains. It is assumed that $Y$ follows a normal distribution $\mathscr{N}\left(110 ; \sigma^{2}\right)$, with mean $\mu = 110$ and standard deviation $\sigma$.
An ice cream is considered marketable when the mass of ice cream it contains belongs to the interval $[104; 116]$.
Determine an approximate value to $10^{-1}$ of the parameter $\sigma$ such that the probability of the event ``the ice cream is marketable'' is equal to 0.98.

We assume in this part that the student uses the bicycle to go to his school. When he uses the bicycle, his travel time, expressed in minutes, between his home and his school is modeled by a random variable $T$ which follows a normal distribution with mean $\mu = 17$ and standard deviation $\sigma = 1.2$.

1. Determine the probability that the student takes between 15 and 20 minutes to get to his school.
2. He leaves his home by bicycle at 7:40 a.m. What is the probability that he is late for school?
3. The student leaves by bicycle. Before what time must he leave to arrive on time at school with a probability of 0.9? Round the result to the nearest minute.

When the student uses the bus, his travel time, expressed in minutes, between his home and his school is modeled by a random variable $T'$ which follows a normal distribution with mean $\mu' = 15$ and standard deviation $\sigma'$. We know that the probability that it takes him more than 20 minutes to get to his school by bus is 0.05. We denote by $Z'$ the random variable equal to $\frac{T' - 15}{\sigma'}$

1. What distribution does the random variable $Z'$ follow?
2. Determine an approximate value to 0.01 of the standard deviation $\sigma'$ of the random variable $T'$.

Question 4

This hypermarket sells baguettes of bread whose mass, expressed in grams, is a random variable that follows a normal distribution with mean 200 g. The probability that the mass of a baguette is between 184 g and 216 g is equal to 0.954. The probability that a baguette chosen at random has a mass less than 192 g has a value rounded to the nearest hundredth of: a. 0.16 b. 0.32 c. 0.84 d. 0.48

A factory produces mineral water in bottles. When the calcium level in a bottle is less than $6.5 \mathrm { mg }$ per litre, the water in that bottle is said to be very low in calcium.
Let $X$ be the random variable that, for each bottle randomly selected from the daily production of source A, associates the calcium level of the water it contains. We assume that $X$ follows a normal distribution with mean 8 and standard deviation 1.6. Let $Y$ be the random variable that, for each bottle randomly selected from the daily production of source B, associates the calcium level it contains. We assume that $Y$ follows a normal distribution with mean 9 and standard deviation $\sigma$.

1. Determine the probability that the calcium level measured in a bottle randomly taken from the daily production of source A is between $6.4 \mathrm { mg }$ and $9.6 \mathrm { mg }$.
2. Calculate the probability $p ( X \leqslant 6.5 )$.
3. Determine $\sigma$ knowing that the probability that a bottle randomly selected from the daily production of source B contains water that is very low in calcium is 0.1.

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 $P(725 \leqslant X \leqslant 775)$.

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 $\sigma = 25$.
The store manager wants to know the number $n$ of budget padlocks he must have in stock at the beginning of the month, so that the probability of running out of stock during the month is less than 0.05. The stock is not replenished during the month.
Determine the smallest integer value of $n$ satisfying this condition.

A company manufactures chocolate tablets of 100 grams. The quality control department performs several types of control.

Part A Control before market release

A chocolate tablet must weigh 100 grams with a tolerance of two grams more or less. It is therefore put on the market if its mass is between 98 and 102 grams. The mass (expressed in grams) of a chocolate tablet can be modelled by a random variable $X$ following the normal distribution with mean $\mu = 100$ and standard deviation $\sigma = 1$. The adjustment of the manufacturing chain machines allows us to modify the value of $\sigma$.

1. Calculate the probability of the event $M$ : ``the tablet is put on the market''.
2. We wish to modify the adjustment of the machines so that the probability of this event reaches 0.97. Determine the value of $\sigma$ so that the probability of the event ``the tablet is put on the market'' equals 0.97.

Part B Control upon reception

The department controls the quality of cocoa beans delivered by producers. One of the quality criteria is the moisture content which must be $7\%$. The bean is then said to be compliant. The company has three different suppliers: the first supplier provides half of the bean stock, the second $30\%$ and the last provides $20\%$ of the stock. For the first, $98\%$ of its production respects the moisture content; for the second, which is somewhat cheaper, $90\%$ of its production is compliant, and the third supplies $20\%$ of non-compliant beans. We randomly choose a bean from the received stock. We denote $F _ { i }$ the event ``the bean comes from supplier $i$'', for $i$ taking the values 1, 2 or 3, and $C$ the event ``the bean is compliant''.

1. Determine the probability that the bean comes from supplier 1, given that it is compliant.

In July 2014, the health surveillance institute of an island published that $15\%$ of the population is affected by the virus. To verify whether the actual proportion is higher, a sample of 1000 people chosen at random from this island is studied. The population is large enough to consider that such a sample results from draws with replacement.
We denote by $X$ the random variable which, for any sample of 1000 people chosen at random, corresponds to the number of people affected by the virus and by $F$ the random variable giving the associated frequency.

1. a. Under the hypothesis $p = 0.15$, determine the distribution of $X$. b. In a sample of 1000 people chosen at random from the island, 197 people affected by the virus are counted. What conclusion can be drawn from this observation about the figure of $15\%$ published by the health surveillance institute? Justify. (You may use the calculation of a fluctuation interval at the $95\%$ threshold.)
2. We now consider that the value of $p$ is unknown. Using the sample from question 1.b., propose a confidence interval for the value of $p$, at the $95\%$ confidence level.

The incubation time, expressed in hours, of the virus can be modeled by a random variable $T$ following a normal distribution with standard deviation $\sigma = 10$. We wish to determine its mean $\mu$.

1. a. Conjecture, using the graph of the probability density function, an approximate value of $\mu$. b. We are given $P(T < 110) = 0.18$. Shade on the graph a region whose area corresponds to the given probability.
2. We denote by $T'$ the random variable equal to $\frac{T - \mu}{10}$. a. What distribution does the random variable $T'$ follow? b. Determine an approximate value to the nearest unit of the mean $\mu$ of the random variable $T$ and verify the conjecture from question 1.

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 a country, the height in centimetres of women aged 18 to 65 can be modelled by a random variable $X _ { 1 }$ following a normal distribution with mean $\mu _ { 1 } = 165 \mathrm {~cm}$ and standard deviation $\sigma _ { 1 } = 6 \mathrm {~cm}$, and that of men aged 18 to 65 by a random variable $X _ { 2 }$ following a normal distribution with mean $\mu _ { 2 } = 175 \mathrm {~cm}$ and standard deviation $\sigma _ { 2 } = 11 \mathrm {~cm}$. In this exercise all results should be rounded to $10 ^ { - 2 }$.

1. What is the probability that a woman chosen at random in this country measures between 1.53 metres and 1.77 metres?
2. a. Determine the probability that a man chosen at random in this country measures more than 1.70 metres. b. Furthermore, it is known that in this country women represent $52 \%$ of the population of people aged between 18 and 65. A person aged between 18 and 65 is chosen at random. They measure more than $1.70 \mathrm {~m}$. What is the probability that this person is a woman?

Exercise 3 -- Common to all candidates

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 $P(X \leqslant 64) = 0.16$.

1. a. By exploiting the graph, determine $P(64 \leqslant X \leqslant 104)$. b. What approximate integer value of $\sigma$ can we propose?
2. 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 $10^{-3}$.
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.

1. 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 $10^{-3}$.
2. 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.

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.

Statistical studies have made it possible to model the weekly time, in hours, of internet connection for young people in France aged 16 to 24 years by a random variable $T$ following a normal distribution with mean $\mu = 13.9$ and standard deviation $\sigma$.

1. We know that $p ( T \geqslant 22 ) = 0.023$.
   By exploiting this information: a. shade on the graph provided in the appendix, two distinct regions whose area is equal to 0.023; b. determine $P ( 5.8 \leqslant T \leqslant 22 )$. Justify the result. Show that an approximate value of $\sigma$ to one decimal place is 4.1.
2. A young person in France is chosen at random.
   Determine the probability that they are connected to the internet for more than 18 hours per week. Round to the nearest hundredth.

The company ``Bonne Mamie'' uses a machine to fill jam jars on a production line. We denote by $X$ the random variable that associates to each jar of jam produced the mass of jam it contains, expressed in grams. In the case where the machine is correctly adjusted, we admit that $X$ follows a normal distribution with mean $\mu = 125$ and standard deviation $\sigma$.

1. a. For any positive real number $t$, determine a relationship between $$P ( X \leqslant 125 - t ) \text { and } P ( X \geqslant 125 + t ) .$$ b. We know that $2.3\%$ of the jam jars contain less than 121 grams of jam. Using the previous relationship, determine $$P ( 121 \leqslant X \leqslant 129 ) .$$
2. Determine a value rounded to the nearest unit of $\sigma$ such that $$P ( 123 \leqslant X \leqslant 127 ) = 0.68 .$$

In the rest of the exercise, we assume that $\boldsymbol { \sigma } = \mathbf { 2 }$.

1. We estimate that a jar of jam is compliant when the mass of jam it contains is between 120 and 130 grams. a. We randomly choose a jar of jam from the production. Determine the probability that this jar is compliant. The result will be given rounded to $10 ^ { - 4 }$. b. We randomly choose a jar from those with a jam mass less than 130 grams. What is the probability that this jar is not compliant? The result will be given rounded to $10 ^ { - 4 }$.
2. We admit that the probability, rounded to $10 ^ { - 3 }$, that a jar of jam is compliant is 0.988. We randomly choose 900 jars from the production. We observe that 871 of these jars are compliant. At the 95\% threshold, can we reject the following hypothesis: ``The machine is correctly adjusted''?

Exercise 4 — Candidates who have NOT followed the specialization course
For each of the following statements, say whether it is true or false by justifying the answer. One point is awarded for each correct justified answer. An unjustified answer will not be taken into account and the absence of an answer is not penalized.

• In the diagram below, the density curve of a random variable $X$ following a normal distribution with mean $\mu = 20$ is represented. The probability that the random variable $X$ is between 20 and 21.6 is equal to 0.34.

Statement 1: The probability that the random variable $X$ belongs to the interval $[23.2; + \infty [$ is approximately 0.046.

• Let $z$ be a complex number different from 2. We set:

$$Z = \frac { \mathrm { i } z } { z - 2 }$$
Statement 2: The set of points in the complex plane with affixe $z$ such that $| Z | = 1$ is a line passing through point $\mathrm { A } ( 1 ; 0 )$. Statement 3: $Z$ is a pure imaginary number if and only if $z$ is real.

• Let $f$ be the function defined on $\mathbb { R }$ by:

$$f ( x ) = \frac { 3 } { 4 + 6 \mathrm { e } ^ { - 2 x } }$$
Statement 4: The equation $f ( x ) = 0.5$ has a unique solution on $\mathbb { R }$. Statement 5: The following algorithm displays as output the value 0.54.

\begin{tabular}{l} Variables: Initialization:

Processing:

Output:

&

$X$ and $Y$ are real numbers

$X$ takes the value 0

$Y$ takes the value $\frac { 3 } { 10 }$

While $Y < 0.5$

$X$ takes the value $X + 0.01$

$Y$ takes the value $\frac { 3 } { 4 + 6 \mathrm { e } ^ { - 2 X } }$

End While

Display $X$

\hline \end{tabular}

Part A
As part of its activity, a company regularly receives quotation requests. The amounts of these quotations are calculated by its secretariat. A statistical study over the past year leads to modelling the amount of quotations by a random variable $X$ which follows the normal distribution with mean $\mu = 2900$ euros and standard deviation $\sigma = 1250$ euros.

1. If a quotation request received by the company is chosen at random, what is the probability that the quotation amount exceeds 4000 euros?
2. In order to improve the profitability of its activity, the entrepreneur decides not to follow up on $10\%$ of requests. He discards those with the lowest quotation amounts. What must be the minimum amount of a requested quotation for it to be taken into account? Give this amount to the nearest euro.

We study the production of a factory that manufactures sweets, packaged in bags. A bag is chosen at random from daily production. The mass of this bag, expressed in grams, is modelled by a random variable $X$ which follows a normal distribution with mean $\mu = 175$. Furthermore, statistical observation has shown that $2\%$ of bags have a mass less than or equal to 170 g, which is expressed in the model considered by: $P ( X \leqslant 170 ) = 0.02$.
What is the probability, rounded to the nearest hundredth, of the event ``the mass of the bag is between 170 and 180 grams''?
Answer a: 0.04 Answer b: 0.96 Answer c: 0.98 Answer d: We cannot answer because data is missing.