Exercise 1 (5 points) -- Common to all candidates
In this exercise, probabilities should be rounded to the nearest hundredth.
Part A
A wholesaler buys boxes of green tea from two suppliers. He buys 80\% of his boxes from supplier A and 20\% from supplier B. 10\% of the boxes from supplier A show traces of pesticides and 20\% of those from supplier B also show traces of pesticides.
A box is randomly selected from the wholesaler's stock and the following events are considered: --- event $A$: ``the box comes from supplier A''; --- event $B$: ``the box comes from supplier B''; --- event $S$: ``the box shows traces of pesticides''.

1. Translate the statement in the form of a weighted tree diagram.
2. a. What is the probability of event $B \cap \bar{S}$? b. Justify that the probability that the selected box shows no traces of pesticides is equal to 0.88.
3. It is observed that the selected box shows traces of pesticides. What is the probability that this box comes from supplier B?

Part B
The manager of a tea salon buys 10 boxes from the above wholesaler. It is assumed that the latter's stock is sufficiently large to model this situation by random selection of 10 boxes with replacement. Consider the random variable $X$ which associates with this sample of 10 boxes the number of boxes without traces of pesticides.

1. Justify that the random variable $X$ follows a binomial distribution and specify its parameters.
2. Calculate the probability that all 10 boxes are free of pesticide traces.
3. Calculate the probability that at least 8 boxes show no traces of pesticides.

Part C
For advertising purposes, the wholesaler displays on his leaflets: ``88\% of our tea is guaranteed free of pesticide traces''.
An inspector from the fraud prevention unit wishes to study the validity of this claim. To this end, he randomly selects 50 boxes from the wholesaler's stock and finds 12 with traces of pesticides.
It is assumed that in the wholesaler's stock, the proportion of boxes without traces of pesticides is indeed equal to 0.88. Let $F$ be the random variable which, for any sample of 50 boxes, associates the frequency of boxes containing no traces of pesticides.

1. Give the asymptotic confidence interval for the random variable $F$ at the 95\% confidence level.
2. Can the fraud prevention inspector decide, at the 95\% confidence level, that the advertisement is misleading?

With three identical valves $V_1, V_2$ and $V_3$, we manufacture a hydraulic circuit. The circuit is operational if $V_1$ is operational or if $V_2$ and $V_3$ are simultaneously operational.
We treat as a random experiment the fact that each valve is or is not operational after 6000 hours. We denote:

• $F_1$ the event: ``valve $V_1$ is operational after 6000 hours''.
• $F_2$ the event: ``valve $V_2$ is operational after 6000 hours''.
• $F_3$ the event: ``valve $V_3$ is operational after 6000 hours''.
• $E$: the event: ``the circuit is operational after 6000 hours''.

We assume that the events $F_1, F_2$ and $F_3$ are pairwise independent and each have probability equal to 0.3.

1. The probability tree shown represents part of the situation. Reproduce this tree and place the probabilities on the branches.
2. Prove that $P(E) = 0.363$.
3. Given that the circuit is operational after 6000 hours, calculate the probability that valve $V_1$ is operational at that time. Round to the nearest thousandth.

A garden centre sells young tree saplings that come from three horticulturists: $35 \%$ of the plants come from horticulturist $\mathrm { H } _ { 1 } , 25 \%$ from horticulturist $\mathrm { H } _ { 2 }$ and the rest from horticulturist $\mathrm { H } _ { 3 }$. Each horticulturist supplies two categories of trees: conifers and deciduous trees. The delivery from horticulturist $\mathrm { H } _ { 1 }$ contains $80 \%$ conifers while that from horticulturist $\mathrm { H } _ { 2 }$ contains only $50 \%$ and that from horticulturist $\mathrm { H } _ { 3 }$ only $30 \%$.
We consider the following events: $H _ { 1 }$ : ``the tree chosen was purchased from horticulturist $\mathrm { H } _ { 1 }$'', $H _ { 2 }$ : ``the tree chosen was purchased from horticulturist $\mathrm { H } _ { 2 }$'', $H _ { 3 }$ : ``the tree chosen was purchased from horticulturist $\mathrm { H } _ { 3 }$'', $C$ : ``the tree chosen is a conifer'', $F$ : ``the tree chosen is a deciduous tree''.

1. The garden centre manager chooses a tree at random from his stock.
   1. [a.] Construct a probability tree representing the situation.
   2. [b.] Calculate the probability that the tree chosen is a conifer purchased from horticulturist $\mathrm { H } _ { 3 }$.
   3. [c.] Justify that the probability of event $C$ is equal to 0.525.
   4. [d.] The tree chosen is a conifer. What is the probability that it was purchased from horticulturist $\mathrm { H } _ { 1 }$? Round to $10 ^ { - 3 }$.
2. A random sample of 10 trees is chosen from the stock of this garden centre. We assume that this stock is large enough that this choice can be treated as sampling with replacement of 10 trees from the stock. Let $X$ be the random variable that gives the number of conifers in the chosen sample.
   1. [a.] Justify that $X$ follows a binomial distribution and specify its parameters.
   2. [b.] What is the probability that the sample contains exactly 5 conifers? Round to $10 ^ { - 3 }$.
   3. [c.] What is the probability that this sample contains at least two deciduous trees? Round to $10 ^ { - 3 }$.

In a factory, two machines A and B are used to manufacture parts.
Machine A ensures $40\%$ of production and machine B ensures $60\%$. It is estimated that $10\%$ of parts from machine A have a defect and that $9\%$ of parts from machine B have a defect.
A part is chosen at random and we consider the following events:

• $A$: ``The part is produced by machine A''
• $B$: ``The part is produced by machine B''
• $D$: ``The part has a defect''
• $\bar{D}$: the opposite event of event $D$.

1. a. Translate the situation using a probability tree. b. Calculate the probability that the chosen part has a defect and was manufactured by machine A. c. Prove that the probability $P(D)$ of event $D$ is equal to 0.094. d. It is observed that the chosen part has a defect. What is the probability that this part comes from machine A?
2. It is estimated that machine A is properly adjusted if $90\%$ of the parts it manufactures are conforming. It is decided to check this machine by examining $n$ parts chosen at random ($n$ natural integer) from the production of machine A. These $n$ draws are treated as successive independent draws with replacement. We denote by $X_n$ the number of parts that are conforming in the sample of $n$ parts, and $F_n = \dfrac{X_n}{n}$ the corresponding proportion. a. Justify that the random variable $X_n$ follows a binomial distribution and specify its parameters. b. In this question, we take $n = 150$. Determine the asymptotic fluctuation interval $I$ at the $95\%$ threshold of the random variable $F_{150}$. c. A quality test counts 21 non-conforming parts in a sample of 150 parts produced. Does this call into question the adjustment of the machine? Justify the answer.

Thomas owns an MP3 player on which he has stored several thousand musical pieces. The set of musical pieces he owns is divided into three distinct genres according to the following distribution: 30\% classical music, 45\% variety, the rest being jazz. Thomas used two encoding qualities to store his musical pieces: high quality encoding and standard encoding. We know that:

• $\frac { 5 } { 6 }$ of the classical music pieces are encoded in high quality.
• $\frac { 5 } { 9 }$ of the variety pieces are encoded in standard quality.

We consider the following events: $C$ : ``The piece heard is a classical music piece''; $V :$ ``The piece heard is a variety piece''; $J$ : ``The piece heard is a jazz piece''; $H$ : ``The piece heard is encoded in high quality''; $S$ : ``The piece heard is encoded in standard quality''.

Part 1

Thomas decides to listen to a piece at random from all the pieces stored on his MP3 using the ``random play'' function. A probability tree may be helpful.

1. What is the probability that it is a classical music piece encoded in high quality?
2. We know that $P ( H ) = \frac { 13 } { 20 }$. a. Are the events $C$ and $H$ independent? b. Calculate $P ( J \cap H )$ and $P _ { J } ( H )$.

Part 2

During a long train journey, Thomas listens to, using the ``random play'' function of his MP3, 60 musical pieces.

1. Determine the asymptotic fluctuation interval at the 95\% threshold of the proportion of classical music pieces in a sample of size 60.
2. Thomas counted that he had listened to 12 classical music pieces during his journey. Can we think that the ``random play'' function of Thomas's MP3 player is defective?

Part 3

Consider the random variable $X$ which, to each song stored on the MP3 player, associates its duration expressed in seconds, and we establish that $X$ follows the normal distribution with mean 200 and standard deviation 20.
We listen to a musical piece at random.

1. Give an approximate value to $10 ^ { - 3 }$ of $P ( 180 \leqslant X \leqslant 220 )$.
2. Give an approximate value to $10 ^ { - 3 }$ of the probability that the piece heard lasts more than 4 minutes.

The company produces $40 \%$ of small-sized footballs and $60 \%$ of standard-sized footballs. It is admitted that $2 \%$ of small-sized footballs and $5 \%$ of standard-sized footballs do not comply with regulations. A football is chosen at random in the company.
Consider the events: $A$ : ``the football is small-sized'', $B$ : ``the football is standard-sized'', $C$ : ``the football complies with regulations'' and $\bar { C }$, the opposite event of C.

1. Represent this random experiment using a probability tree.
2. Calculate the probability that the football is small-sized and complies with regulations.
3. Show that the probability of event $C$ is equal to 0.962.
4. The football chosen does not comply with regulations. What is the probability that this football is small-sized? Round the result to $10 ^ { - 3 }$.

Exercise 1 (5 points)

Part A

An oyster farmer raises two species of oysters: ``the flat'' and ``the Japanese''. Each year, flat oysters represent $15 \%$ of his production. Oysters are said to be of size $\mathrm { n } ^ { \circ } 3$ when their mass is between 66 g and 85 g. Only $10 \%$ of flat oysters are of size $\mathrm { n } ^ { \circ } 3$, whereas $80 \%$ of Japanese oysters are.

1. The health service randomly selects an oyster from the oyster farmer's production. We assume that all oysters have an equal chance of being selected. We consider the following events:
   • J: ``the selected oyster is a Japanese oyster'',
   • C: ``the selected oyster is of size $\mathrm { n } ^ { \circ } 3$''.
   a. Construct a complete weighted tree representing the situation. b. Calculate the probability that the selected oyster is a flat oyster of size $\mathrm { n } ^ { \mathrm { o } } 3$. c. Justify that the probability of obtaining an oyster of size $\mathrm { n } ^ { \circ } 3$ is 0.695. d. The health service selected an oyster of size $\mathrm { n } ^ { \circ } 3$. What is the probability that it is a flat oyster?
   
2. The mass of an oyster can be modeled by a random variable $X$ following the normal distribution with mean $\mu = 90$ and standard deviation $\sigma = 2$. a. Give the probability that the oyster selected from the oyster farmer's production has a mass between 87 g and 89 g. b. Give $\mathrm { P } ( \mathrm { X } \geqslant 91 )$.

Part B

This oyster farmer claims that $60 \%$ of his oysters have a mass greater than 91 g.
A restaurant owner would like to buy a large quantity of oysters but would first like to verify the oyster farmer's claim.
The restaurant owner purchases 10 dozen oysters from this oyster farmer, which we will consider as a sample of 120 oysters drawn at random. His production is large enough that we can treat it as sampling with replacement. He observes that 65 of these oysters have a mass greater than 91 g.

1. Let F be the random variable that associates to any sample of 120 oysters the frequency of those with a mass greater than 91 g. After verifying the conditions of application, give an asymptotic confidence interval at the $95 \%$ level for the random variable F.
2. What can the restaurant owner think of the oyster farmer's claim?

Question 1

In a hypermarket, $75\%$ of customers are women. One woman in five buys an item from the DIY section, whereas seven men in ten do so.
A person, chosen at random, has made a purchase from the DIY section. The probability that this person is a woman has a value rounded to the nearest thousandth of: a. 0.750 b. 0.150 c. 0.462 d. 0.700

Parts A and B can be treated independently.

Part A

A pharmaceutical laboratory offers screening tests for various diseases. Its communications department highlights the following characteristics:

• the probability that a sick person tests positive is 0.99;
• the probability that a healthy person tests positive is 0.001.

1. For a disease that has just appeared, the laboratory develops a new test. A statistical study makes it possible to estimate that the percentage of sick people among the population of a metropolis is equal to $0.1 \%$. A person is chosen at random from this population and undergoes the test. We denote by $M$ the event ``the chosen person is sick'' and $T$ the event ``the test is positive''. a. Translate the statement in the form of a weighted tree. b. Prove that the probability $p ( T )$ of event $T$ is equal to $$1.989 \times 10 ^ { - 3 } .$$ c. Is the following statement true or false? Justify your answer. Statement: ``If the test is positive, there is less than one chance in two that the person is sick''.
2. The laboratory decides to market a test as soon as the probability that a person who tests positive is sick is greater than or equal to 0.95. We denote by $x$ the proportion of people affected by a certain disease in the population. From what value of $x$ does the laboratory market the corresponding test?

Part B

The laboratory's production line manufactures, in very large quantities, tablets of a medicine.

1. A tablet is compliant if its mass is between 890 and 920 mg. We assume that the mass in milligrams of a tablet taken at random from production can be modeled by a random variable $X$ that follows the normal distribution $\mathscr { N } \left( \mu , \sigma ^ { 2 } \right)$, with mean $\mu = 900$ and standard deviation $\sigma = 7$. a. Calculate the probability that a tablet drawn at random is compliant. Round to $10 ^ { - 2 }$. b. Determine the positive integer $h$ such that $P ( 900 - h \leqslant X \leqslant 900 + h ) \approx 0.99$ to within $10 ^ { - 3 }$.
2. The production line has been adjusted to obtain at least $97 \%$ compliant tablets. To evaluate the effectiveness of the adjustments, a check is performed by taking a sample of 1000 tablets from production. The size of the production is assumed to be large enough that this sample can be treated as 1000 successive draws with replacement. The check made it possible to count 53 non-compliant tablets in the sample taken. Does this check call into question the adjustments made by the laboratory? An asymptotic fluctuation interval at the $95 \%$ threshold can be used.

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.

1. 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.''
2. 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.''
3. 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.''
4. 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.''

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.
The mineral water comes from two sources, noted ``source A'' and ``source B''. The probability that water from a bottle randomly selected from the daily production of source A is very low in calcium is 0.17. The probability that water from a bottle randomly selected from the daily production of source B is very low in calcium is 0.10. Source A supplies $70\%$ of the total daily production of water bottles and source B supplies the rest of this production. A water bottle is randomly selected from the total daily production. We consider the following events: A: ``The water bottle comes from source A'' B: ``The water bottle comes from source B'' $S$: ``The water contained in the water bottle is very low in calcium''.

1. Determine the probability of event $A \cap S$.
2. Show that the probability of event $S$ equals 0.149.
3. Calculate the probability that the water contained in a bottle comes from source A given that it is very low in calcium.
4. The day after heavy rain, the factory takes a sample of 1000 bottles from source A. Among these bottles, 211 contain water that is very low in calcium. Give an interval to estimate at the $95\%$ confidence level the proportion of bottles containing water that is very low in calcium in the entire production of source A after this weather event.

It is now assumed that, in the store:

• $80\%$ of the padlocks offered for sale are budget models, the others are high-end;
• $3\%$ of high-end padlocks are defective;
• $7\%$ of padlocks are defective.

A padlock is randomly selected from the store. We denote:

• $p$ the probability that a budget padlock is defective;
• $H$ the event: ``the selected padlock is high-end'';
• $D$ the event: ``the selected padlock is defective''.

1. Represent the situation using a probability tree.
2. Express $P(D)$ as a function of $p$. Deduce the value of the real number $p$.

Is the result obtained consistent with that of question A-2?
3. The selected padlock is in good condition. Determine the probability that it is a high-end padlock.

A chain of stores wishes to build customer loyalty by offering gift vouchers to its privileged customers. Each of them receives a gift voucher of green or red colour on which an amount is written. Gift vouchers are distributed so as to have, in each store, one quarter of red vouchers and three quarters of green vouchers. Green gift vouchers take the value of 30 euros with a probability equal to 0.067 or values between 0 and 15 euros with unspecified probabilities here. Similarly, red gift vouchers take the values 30 or 100 euros with probabilities respectively equal to 0.015 and 0.010 or values between 10 and 20 euros with unspecified probabilities here.
Calculate the probability of having a gift voucher with a value greater than or equal to 30 euros knowing that it is red.

A chain of stores wishes to build customer loyalty by offering gift vouchers to its privileged customers. Each of them receives a gift voucher of green or red colour on which an amount is written. Gift vouchers are distributed so as to have, in each store, one quarter of red vouchers and three quarters of green vouchers. Green gift vouchers take the value of 30 euros with a probability equal to 0.067 or values between 0 and 15 euros with unspecified probabilities here. Similarly, red gift vouchers take the values 30 or 100 euros with probabilities respectively equal to 0.015 and 0.010 or values between 10 and 20 euros with unspecified probabilities here.
Show that an approximate value to $10 ^ { - 3 }$ near of the probability of having a gift voucher with a value greater than or equal to 30 euros is equal to 0.057. For the following question, this value is used.

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.

Chikungunya is a viral disease transmitted from one human to another by bites of infected female mosquitoes. A test has been developed for the screening of this virus. The laboratory manufacturing this test provides the following characteristics:

• the probability that a person affected by the virus has a positive test is 0.98;
• the probability that a person not affected by the virus has a positive test is 0.01.

An individual is chosen at random from a target population. We call:

• $M$ the event: "The chosen individual is affected by chikungunya"
• $T$ the event: "The test of the chosen individual is positive"

We denote by $\bar{M}$ (respectively $\bar{T}$) the opposite event of the event $M$ (respectively $T$). We denote by $p$ $(0 \leqslant p \leqslant 1)$ the proportion of people affected by the disease in the target population.

1. a. Copy and complete the probability tree. b. Express $P(M \cap T)$, $P(\bar{M} \cap T)$ then $P(T)$ as a function of $p$.
2. a. Prove that the probability of $M$ given $T$ is given by the function $f$ defined on $[0;1]$ by: $$f(p) = \frac{98p}{97p + 1}$$ b. Study the variations of the function $f$.
3. We consider that the test is reliable when the probability that a person with a positive test is actually affected by chikungunya is greater than 0.95. Using the results of question 2., from what proportion $p$ of sick people in the population is the test reliable?

Candidates who have not followed the specialization course
In preparation for an election between two candidates A and B, a polling institute collects the voting intentions of future voters. Among the 1200 people who responded to the survey, $47\%$ state they want to vote for candidate A and the others for candidate B.
Given the profile of the candidates, the polling institute estimates that $10\%$ of people declaring they want to vote for candidate A are not telling the truth and actually vote for candidate B, while $20\%$ of people declaring they want to vote for candidate B are not telling the truth and actually vote for candidate A.
We randomly choose a person who responded to the survey and we denote:

• A the event ``The person interviewed states they want to vote for candidate A'';
• $B$ the event ``The person interviewed states they want to vote for candidate B'';
• $V$ the event ``The person interviewed is telling the truth''.

1. Construct a probability tree representing the situation.
2. a) Calculate the probability that the person interviewed is telling the truth. b) Given that the person interviewed is telling the truth, calculate the probability that they state they want to vote for candidate A.
3. Prove that the probability that the chosen person actually votes for candidate A is 0.529.
4. The polling institute then publishes the following results:
   \begin{displayquote} $52.9\%$ of voters* would vote for candidate A. *estimate after adjustment, based on a survey of a representative sample of 1200 people. \end{displayquote}
   At the 95\% confidence level, can candidate A believe in their victory?
5. To conduct this survey, the institute conducted a telephone survey at a rate of 10 calls per half-hour. The probability that a person contacted agrees to respond to this survey is 0.4. The polling institute wishes to obtain a sample of 1200 responses. What average time, expressed in hours, should the institute plan to achieve this objective?

Part B
This same entrepreneur decides to install anti-spam software. This software detects unwanted messages called spam (malicious messages, advertisements, etc.) and moves them to a file called the ``spam folder''. The manufacturer claims that $95\%$ of spam messages are moved. For his part, the entrepreneur knows that $60\%$ of the messages he receives are spam. After installing the software, he observes that $58.6\%$ of messages are moved to the spam folder. For a message chosen at random, we consider the following events:

• $D$: ``the message is moved'';
• $S$: ``the message is spam''.

1. Calculate $P ( S \cap D )$.
2. A message that is not spam is chosen at random. Show that the probability that it is moved equals 0.04.
3. A message that is not moved is chosen at random. What is the probability that this message is spam?
4. For the software chosen by the company, the manufacturer estimates that $2.7\%$ of messages moved to the spam folder are reliable messages. In order to test the software's effectiveness, the secretariat takes the trouble to count the number of reliable messages among the moved messages. It finds 13 reliable messages among the 231 messages moved during one week. Do these results call into question the manufacturer's claim?

Sofia wishes to go to the cinema. She can go by bike or by bus.

Part A: Using the bus

We assume in this part that Sofia uses the bus to go to the cinema. The duration of the journey between her home and the cinema (expressed in minutes) is modelled by the random variable $T _ { B }$ which follows the uniform distribution on [12; 15].

1. Prove that the probability that Sofia takes between 12 and 14 minutes is $\frac { 2 } { 3 }$.
2. Give the average duration of the journey.

Part B: Using her bike

We now assume that Sofia chooses to use her bike. The duration of the journey (expressed in minutes) is modelled by the random variable $T _ { v }$ which follows the normal distribution with mean $\mu = 14$ and standard deviation $\sigma = 1,5$.

1. What is the probability that Sofia takes less than 14 minutes to go to the cinema? What is the probability that Sofia takes between 12 and 14 minutes to go to the cinema? Round the result to $10 ^ { - 3 }$.

Part C: Playing with dice

Sofia is hesitating between the bus and the bike. She decides to roll a fair 6-sided die. If she gets 1 or 2, she takes the bus, otherwise she takes her bike. We denote:

• $B$ the event ``Sofia takes the bus'';
• $V$ the event ``Sofia takes her bike'';
• C the event ``Sofia takes between 12 and 14 minutes to go to the cinema''.

1. Prove that the probability, rounded to $10 ^ { - 2 }$, that Sofia takes between 12 and 14 minutes is 0.49.
2. Given that Sofia took between 12 and 14 minutes to go to the cinema, what is the probability, rounded to $10 ^ { - 2 }$, that she used the bus?

Part B - Reaching an operator
If the waiting time before reaching an operator exceeds 5 minutes, the call automatically ends. Otherwise, the caller reaches an operator. We randomly choose a customer who calls the assistance line. We assume that the probability that the call comes from an Internet customer is 0.7. Furthermore, according to Part A, we take the following data:

• If the call comes from an Internet customer then the probability of reaching an operator is equal to 0.95.
• If the call comes from a mobile customer then the probability of reaching an operator is equal to 0.87.

1. Determine the probability that the customer reaches an operator.
2. A customer complains that their call ended after 5 minutes of waiting without reaching an operator. Is it more likely that this is an Internet customer or a mobile customer?

Machine A produces one third of the factory's sweets. The rest of production is ensured by machine B. When produced by machine B, the probability that a randomly selected sweet is deformed is equal to 0.02. In a quality control test, a sweet is randomly selected from the entire production. It is deformed.
What is the probability, rounded to the nearest hundredth, that it was produced by machine B?
Answer a: 0.02 Answer b: 0.67 Answer c: 0.44 Answer d: 0.01

Exercise 1

4 points

Common to all candidates

Parts $\mathrm { A } , \mathrm { B }$ and C can be treated independently. Throughout the exercise, results should be rounded, if necessary, to the nearest thousandth.

Part A

A merchant receives the results of a market study on consumer habits in France. According to this study:

• $54 \%$ of consumers prefer products of French manufacture;
• $65 \%$ of consumers regularly buy products from organic agriculture, and among them $72 \%$ prefer products of French manufacture.

A consumer is chosen at random. The following events are considered:

• B: ``a consumer regularly buys products from organic agriculture'';
• $F$ : ``a consumer prefers products of French manufacture''.

We denote $P ( A )$ the probability of event $A$ and $P _ { C } ( A )$ the probability of $A$ given $C$.

1. Justify that $P ( \bar { B } \cap F ) = 0.072$.
2. Calculate $P _ { F } ( \bar { B } )$.
3. A consumer is chosen who does not regularly buy products from organic agriculture. What is the probability that he prefers products of French manufacture?

Part B

The merchant is interested in the quantity in kilograms of organic flour sold each month at retail in his store. This quantity is modeled by a random variable $X$ which follows the normal distribution with mean $\mu = 90$ and standard deviation $\sigma = 2$.

1. At the beginning of each month, the merchant ensures he has 95 kg in stock.

What is the probability that he cannot meet customer demand during the month?
2. Determine an approximate value to the nearest hundredth of the real number $a$ such that $P ( X < a ) = 0.02$.
Interpret the result in the context of the exercise.

Part C

In this market study, it is specified that $46.8 \%$ of consumers in France prefer local products. The merchant observes that among his 2500 customers, 1025 regularly buy local products. Is his customer base representative of consumers in France?

A company packages white sugar from two farms $U$ and $V$ in 1 kg packets. We admit that $3\%$ of the sugar from farm $U$ is extra fine and that $5\%$ of the sugar from farm V is extra fine. A packet of sugar is randomly selected from the company's production and we consider the following events:

• $U$: ``The packet contains sugar from farm U'';
• $V$: ``The packet contains sugar from farm V'';
• $E$: ``The packet bears the label `extra fine' ''.

1. In this question, we admit that the company manufactures $30\%$ of its packets with sugar from farm U and the others with sugar from farm V, without mixing sugars from the two farms. a. What is the probability that the selected packet bears the label ``extra fine''? b. Given that a packet bears the label ``extra fine'', what is the probability that the sugar it contains comes from farm U?
2. The company wishes to modify its supply from the two farms so that among the packets bearing the label ``extra fine'', $30\%$ of them contain sugar from farm U. How should it supply itself from farms U and V? Any working will be valued in this question.

This club makes group orders of bearings for its members from two suppliers A and B.

• Supplier A offers higher prices but the bearings it sells are defect-free with a probability of 0.97.
• Supplier B offers more advantageous prices but its bearings are defective with a probability of 0.05.

A bearing is chosen at random from the club's stock and we consider the events: $A$: ``the bearing comes from supplier A'', $B$: ``the bearing comes from supplier B'', $D$: ``the bearing is defective''.

1. The club buys $40\%$ of its bearings from supplier A and the rest from supplier B. a. Calculate the probability that the bearing comes from supplier A and is defective. b. The bearing is defective. Calculate the probability that it comes from supplier B.
2. If the club wants less than $3.5\%$ of the bearings to be defective, what minimum proportion of bearings should it order from supplier A?

In France, the consumption of organic products has been growing for several years.
In 2017, the country had $52\%$ women. That same year, $92\%$ of French people had already consumed organic products. Furthermore, among consumers of organic products, $55\%$ were women.
We randomly choose a person from the file of French people in 2017. We denote:

• $F$ the event ``the chosen person is a woman'';
• $H$ the event ``the chosen person is a man'';
• $B$ the event ``the chosen person has already consumed organic products''.

1. Translate the numerical data from the statement using events $F$ and $B$.
2. a. Show that $P(F \cap B) = 0{,}506$. b. Deduce the probability that a person consumed organic products in 2017, given that they are a woman.
3. Calculate $P_H(\bar{B})$. Interpret this result in the context of the exercise.