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?

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.

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?

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.

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?

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.

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?

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?

Exercise 4 -- Candidates who have not followed the specialisation course
Part A
During an evening, a television channel broadcast a match. This channel then offered a programme analysing this match. We have the following information:

• $56\%$ of viewers watched the match;
• one quarter of viewers who watched the match also watched the programme;
• $16.2\%$ of viewers watched the programme.

We randomly interview a viewer. We denote the events:

• $M$: ``the viewer watched the match'';
• $E$: ``the viewer watched the programme''.

We denote by $x$ the probability that a viewer watched the programme given that they did not watch the match.

1. Construct a probability tree illustrating the situation.
2. Determine the probability of $M \cap E$.
3. 1. [a.] Verify that $p ( E ) = 0.44 x + 0.14$.
   2. [b.] Deduce the value of $x$.
4. The interviewed viewer did not watch the programme. What is the probability, rounded to $10 ^ { - 2 }$, that they watched the match?

Part B
This institute decides to model the time spent, in hours, by a viewer watching television on the evening of the match, by a random variable $T$ following the normal distribution with mean $\mu = 1.5$ and standard deviation $\sigma = 0.5$.

1. What is the probability, rounded to $10 ^ { - 3 }$, that a viewer spent between one hour and two hours watching television on the evening of the match?
2. Determine the approximation to $10 ^ { - 2 }$ of the real number $t$ such that $P ( T \geqslant t ) = 0.066$. Interpret the result.

Part C
The lifetime of an individual set-top box, expressed in years, is modelled by a random variable denoted $S$ which follows an exponential distribution with parameter $\lambda$ strictly positive. The probability density function of $S$ is the function $f$ defined on $[ 0 ; + \infty [$ by $$f ( x ) = \lambda \mathrm { e } ^ { - \lambda x }$$ The polling institute has observed that one quarter of the set-top boxes have a lifetime between one and two years. The factory that manufactures the set-top boxes claims that their average lifetime is greater than three years. Is the factory's claim correct? The answer must be justified.

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.

Part A

Louise drives to work with her car. Her colleague Zoé does not own a car. Each morning, Louise therefore offers to give Zoé a ride. Whatever Zoé's answer, Louise offers to drive her back in the evening. We consider a given day. We have the following information:

• the probability that Louise drives Zoé in the morning is 0.55;
• if Louise drove Zoé in the morning, the probability that she drives her back in the evening is 0.7;
• if Louise did not drive Zoé in the morning, the probability that she drives her back in the evening is 0.24.

We denote $M$ and $S$ the following events:

• $M$: ``Louise drives Zoé in the morning'';
• S: ``Louise drives Zoé back in the evening''.

1. Construct a probability tree representing the situation.
2. Calculate $P ( M \cap S )$. Translate this result with a sentence.
3. Prove that the probability of event S is equal to 0.493.
4. We know that Louise drove Zoé back in the evening. What is the probability that Louise drove her in the morning?

In a statistics school, after reviewing candidate files, recruitment is done in two ways:

• $10\%$ of candidates are selected based on their file. These candidates must then take an oral examination after which $60\%$ of them are finally admitted to the school.
• Candidates who were not selected based on their file take a written examination after which $20\%$ of them are admitted to the school.

Part 1
A candidate for this recruitment competition is chosen at random. We denote:

• $D$ the event ``the candidate was selected based on their file'';
• $A$ the event ``the candidate was admitted to the school'';
• $\bar{D}$ and $\bar{A}$ the complementary events of events $D$ and $A$ respectively.

1. Represent the situation with a probability tree.
2. Calculate the probability that the candidate is selected based on their file and admitted to the school.
3. Show that the probability of event $A$ is equal to 0.24.
4. A candidate admitted to the school is chosen at random. What is the probability that their file was not selected?

Part 2

1. We assume that the probability for a candidate to be admitted to the school is equal to 0.24.
   We consider a sample of seven candidates chosen at random, treating this choice as a random draw with replacement. We denote by $X$ the random variable counting the candidates admitted to the school among the seven drawn. a. We assume that the random variable $X$ follows a binomial distribution. What are the parameters of this distribution? b. Calculate the probability that only one of the seven candidates drawn is admitted to the school. Give an answer rounded to the nearest hundredth. c. Calculate the probability that at least two of the seven candidates drawn are admitted to this school. Give an answer rounded to the nearest hundredth.
2. A secondary school presents $n$ candidates for recruitment in this school, where $n$ is a non-zero natural number. We assume that the probability for any candidate from the secondary school to be admitted to the school is equal to 0.24 and that the results of the candidates are independent of each other. a. Give the expression, as a function of $n$, of the probability that no candidate from this secondary school is admitted to the school. b. From what value of the integer $n$ is the probability that at least one student from this secondary school is admitted to the school greater than or equal to 0.99?

Every day he works, Paul must go to the station to reach his workplace by train. To do this, he takes his bicycle two times out of three and, if he does not take his bicycle, he takes his car.

1. When he takes his bicycle to reach the station, Paul misses the train only once in 50, whereas when he takes his car to reach the station, Paul misses his train once in 10. We consider a random day on which Paul will be at the station to catch the train that will take him to work. We denote:
   • V the event ``Paul takes his bicycle to reach the station'';
   • R the event ``Paul misses his train''. a. Draw a weighted tree summarizing the situation. b. Show that the probability that Paul misses his train is equal to $\frac { 7 } { 150 }$. c. Paul has missed his train. Determine the exact value of the probability that he took his bicycle to reach the station.
   
   
2. A random month is chosen during which Paul went to the station 20 days to reach his workplace according to the procedures described in the preamble. We assume that, for each of these 20 days, the choice between bicycle and car is independent of the choices on other days. We denote $X$ the random variable giving the number of days Paul takes his bicycle over these 20 days. a. Determine the distribution followed by the random variable $X$. Specify its parameters. b. What is the probability that Paul takes his bicycle exactly 10 days out of these 20 days to reach the station? Round the probability sought to $10 ^ { - 3 }$. c. What is the probability that Paul takes his bicycle at least 10 days out of these 20 days to reach the station? Round the probability sought to $10 ^ { - 3 }$. d. On average, how many days over a randomly chosen period of 20 days to reach the station does Paul take his bicycle? Round the answer to the nearest integer.
   
3. In the case where Paul goes to the station by car, we denote $T$ the random variable giving the travel time needed to reach the station. The duration of the journey is given in minutes, rounded to the minute. The probability distribution of $T$ is given by the table below:

$k$ (in minutes)	10	11	12	13	14	15	16	17	18
$P ( T = k )$	0,14	0,13	0,13	0,12	0,12	0,11	0,10	0,08	0,07

Determine the expected value of the random variable $T$ and interpret this value in the context of the exercise.

Part 1

Julien must take the plane; he planned to take the bus to get to the airport. If he takes the 8 o'clock bus, he is sure to be at the airport in time for his flight. On the other hand, the next bus would not allow him to arrive at the airport in time. Julien left late from his apartment and the probability that he misses his bus is 0.8. If he misses his bus, he goes to the airport by taking a private car company; he then has a probability of 0.5 of being on time at the airport. We denote:

• $B$ the event: ``Julien manages to take his bus'';
• $V$ the event: ``Julien is on time at the airport for his flight''.

1. Give the value of $P _ { B } ( V )$.
2. Represent the situation with a probability tree.
3. Show that $P ( V ) = 0.6$.
4. If Julien is on time at the airport for his flight, what is the probability that he arrived at the airport by bus? Justify.

Part 2

Airlines sell more tickets than there are seats on planes because some passengers do not show up for boarding on the flight they have booked. This practice is called overbooking. Based on statistics from previous flights, the airline estimates that each passenger has a 5\% chance of not showing up for boarding. Consider a flight on a plane with 200 seats for which 206 tickets have been sold. We assume that the presence at boarding of each passenger is independent of other passengers and we call $X$ the random variable that counts the number of passengers showing up for boarding.

1. Justify that $X$ follows a binomial distribution and specify its parameters.
2. On average, how many passengers will show up for boarding?
3. Calculate the probability that 201 passengers show up for boarding. The result should be rounded to $10 ^ { - 3 }$.
4. Calculate $P ( X \leqslant 200 )$, the result should be rounded to $10 ^ { - 3 }$. Interpret this result in the context of the exercise.
5. The airline sells each ticket for 250 euros.

If more than 200 passengers show up for boarding, the airline must refund the plane ticket and pay a penalty of 600 euros to each affected passenger. We call: $Y$ the random variable equal to the number of passengers who cannot board despite having purchased a ticket; $C$ the random variable that totals the revenue of the airline on this flight.
We admit that $Y$ follows the probability distribution given by the following table:

$y _ { i }$	0	1	2	3	4	5	6
$P \left( Y = y _ { i } \right)$	0,94775	0,03063	0,01441	0,00539	0,00151	0,00028

a. Complete the probability distribution given above by calculating $P ( Y = 6 )$. b. Justify that: $C = 51500 - 850 Y$. c. Give the probability distribution of the random variable $C$ in the form of a table. Calculate the expected value of the random variable $C$ to the nearest euro. d. Compare the revenue obtained by selling exactly 200 tickets and the average revenue obtained by practicing overbooking.

Consider a binary communication system transmitting 0s and 1s. Each 0 or 1 is called a bit. Due to interference, there may be transmission errors: a 0 can be received as a 1 and, likewise, a 1 can be received as a 0. For a bit chosen at random in the message, we note the events:

• $E _ { 0 }$: ``the bit sent is a 0'';
• $E _ { 1 }$: ``the bit sent is a 1'';
• $R _ { 0 }$: ``the bit received is a 0''
• $R _ { 1 }$: ``the bit received is a 1''.

We know that: $p \left( E _ { 0 } \right) = 0{,}4 ; \quad p _ { E _ { 0 } } \left( R _ { 1 } \right) = 0{,}01 ; \quad p _ { E _ { 1 } } \left( R _ { 0 } \right) = 0{,}02$. Recall that the conditional probability of $A$ given $B$ is denoted $p _ { B } ( A )$.

1. The probability that the bit sent is a 0 and the bit received is a 0 is equal to: a. 0,99 b. 0,396 c. 0,01 d. 0,4
2. The probability $p \left( R _ { 0 } \right)$ is equal to: a. 0,99 b. 0,02 c. 0,408 d. 0,931
3. A value, approximated to the nearest thousandth, of the probability $p _ { R _ { 1 } } \left( E _ { 0 } \right)$ is equal to: a. 0,004 b. 0,001 c. 0,007 d. 0,010
4. The probability of the event ``there is a transmission error'' is equal to: a. 0,03 b. 0,016 c. 0,16 d. 0,015

A message of length eight bits is called a byte. It is admitted that the probability that a byte is transmitted without error is equal to 0,88.

1. 10 bytes are transmitted successively in an independent manner.
   The probability, to $10 ^ { - 3 }$ near, that exactly 7 bytes are transmitted without error is equal to: a. 0,915 b. 0,109 c. 0,976 d. 0,085
2. 10 bytes are transmitted successively in an independent manner.
   The probability that at least 1 byte is transmitted without error is equal to: a. $1 - 0{,}12 ^ { 10 }$ b. $0{,}12 ^ { 10 }$ c. $0{,}88 ^ { 10 }$ d. $1 - 0{,}88 ^ { 10 }$
3. Let $N$ be a natural integer. $N$ bytes are transmitted successively in an independent manner. Let $N _ { 0 }$ be the largest value of $N$ for which the probability that all $N$ bytes are transmitted without error is greater than or equal to 0,1. We can affirm that: a. $N _ { 0 } = 17$ b. $N _ { 0 } = 18$ c. $N _ { 0 } = 19$ d. $N _ { 0 } = 20$

A hotel located near a prehistoric tourism site offers two visits in the surrounding area, one to a museum and one to a cave.
A study showed that $70\%$ of the hotel's clients visit the museum. Furthermore, among clients visiting the museum, $60\%$ visit the cave. The study also shows that $6\%$ of the hotel's clients make no visits. We randomly question a hotel client and note:

• $M$ the event: ``the client visits the museum'';
• $G$ the event: ``the client visits the cave''.

We denote by $\bar { M }$ the complementary event of $M$, $\bar { G }$ the complementary event of $G$, and for any event $E$, we denote by $p ( E )$ the probability of $E$. Thus, according to the problem statement, we have: $p ( \bar { M } \cap \bar { G } ) = 0.06$.

1. a. Verify that $p _ { \bar { M } } ( \bar { G } ) = 0.2$, where $p _ { \bar { M } } ( \bar { G } )$ denotes the probability that the questioned client does not visit the cave given that he does not visit the museum. b. The weighted tree opposite models the situation. Copy and complete this tree by indicating on each branch the associated probability. c. What is the probability of the event ``the client visits the cave and does not visit the museum''? d. Show that $p ( G ) = 0.66$.
2. The hotel manager claims that among clients who visit the cave, more than half also visit the museum. Is this claim correct?
3. The prices for visits are as follows:
   • museum visit: 12 euros;
   • cave visit: 5 euros.
   We consider the random variable $T$ which models the amount spent by a hotel client for these visits. a. Give the probability distribution of $T$. Present the results in the form of a table. b. Calculate the mathematical expectation of $T$. c. For profitability reasons, the hotel manager estimates that the average amount of visit revenues must be greater than 700 euros per day. Determine the average number of clients per day needed to achieve this objective.
4. To increase revenues, the manager wishes the expectation of the random variable modeling the amount spent by a hotel client for these visits to increase to 15 euros, without changing the museum visit price which remains at 12 euros. What price should be set for the cave visit to achieve this objective? (We will assume that the increase in the cave entrance price does not change the frequency of visits to the two sites).
5. We randomly choose 100 hotel clients, treating this choice as a draw with replacement. What is the probability that at least three-quarters of these clients visited the cave during their stay at the hotel? Give a value of the result to $10 ^ { - 3 }$ near.

Exercise 3 (7 points) The director of a large company proposed a training course to all its employees on the use of new software. This course was followed by $25\%$ of employees.

1. In this company, $52\%$ of employees are women, of whom $40\%$ followed the course.

A random employee of the company is questioned and we consider the events:

• $F$: ``the employee questioned is a woman'',
• $S$: ``the employee questioned followed the course''. $\bar{F}$ and $\bar{S}$ denote respectively the complementary events of events $F$ and $S$. a. Give the probability of event $S$. b. Copy and complete the blanks of the probability tree below on the four indicated branches. c. Demonstrate that the probability that the person questioned is a woman who followed the course is equal to 0.208. d. Given that the person questioned followed the course, what is the probability that it is a woman? e. The director claims that, among the male employees of the company, fewer than $10\%$ followed the course. Justify the director's claim.

1. We denote by $X$ the random variable that associates to a sample of 20 employees of this company chosen at random the number of employees in this sample who followed the course. We assume that the number of employees in the company is sufficiently large to assimilate this choice to sampling with replacement. a. Determine, by justifying, the probability distribution followed by the random variable $X$. b. Determine, to $10^{-3}$ near, the probability that 5 employees in a sample of 20 followed the course. c. The program below, written in Python language, uses the function binomial$(i, n, p)$ created for this purpose which returns the value of the probability $P(X = i)$ in the case where the random variable $X$ follows a binomial distribution with parameters $n$ and $p$. \begin{verbatim} def proba(k) : P=0 for i in range(0,k+1) : P=P+binomiale(i,20,0.25) return P \end{verbatim} Determine, to $10^{-3}$ near, the value returned by this program when proba(5) is entered in the Python console. Interpret this value in the context of the exercise. d. Determine, to $10^{-3}$ near, the probability that at least 6 employees in a sample of 20 followed the course.
2. This question is independent of questions 1 and 2. To encourage employees to follow the course, the company had decided to increase the salaries of employees who followed the course by $5\%$, compared to $2\%$ increase for employees who did not follow the course. What is the average percentage increase in salaries for this company under these conditions?

In an effort to improve its sustainable development policy, a company conducted a statistical survey on its waste production.
In this survey, waste is classified into three categories:

• $69 \%$ of waste is mineral and non-hazardous;
• $28 \%$ of waste is non-mineral and non-hazardous;
• the remaining waste is hazardous waste.

This statistical survey also tells us that:

• $73 \%$ of mineral and non-hazardous waste is recyclable;
• $49 \%$ of non-mineral and non-hazardous waste is recyclable;
• $35 \%$ of hazardous waste is recyclable.

In this company, a piece of waste is randomly selected. We consider the following events:

• $M$ : ``The selected waste is mineral and non-hazardous'';
• N : ``The selected waste is non-mineral and non-hazardous'';
• $D$ : ``The selected waste is hazardous'';
• R: ``The selected waste is recyclable''.

We denote by $\bar{R}$ the complementary event of event $R$.
Part A

1. Copy and complete the probability tree below representing the situation described in the problem.
2. Justify that the probability that the waste is hazardous and recyclable is equal to 0.0105.
3. Determine the probability $P(M \cap \bar{R})$ and interpret the answer obtained in the context of the exercise.
4. Prove that the probability of event $R$ is $P(R) = 0.6514$.
5. Suppose that the selected waste is recyclable. Determine the probability that this waste is non-mineral and non-hazardous. Give the answer rounded to the ten-thousandth.

Part B
We recall that the probability that a randomly selected piece of waste is recyclable is equal to 0.6514.

1. In order to control the quality of collection in the company, a sample of 20 pieces of waste is randomly selected from production. We assume that the stock is sufficiently large to treat the sampling of this sample as drawing with replacement.
   We denote by $X$ the random variable equal to the number of recyclable pieces of waste in this sample. a. We admit that the random variable $X$ follows a binomial distribution. Specify its parameters. b. Give the probability that the sample contains exactly 14 recyclable pieces of waste. Give the answer rounded to the ten-thousandth.
2. In this question, we now select $n$ pieces of waste, where $n$ denotes a strictly positive natural number. a. Give the expression as a function of $n$ of the probability $p_n$ that no piece of waste in this sample is recyclable. b. Determine the value of the natural number $n$ from which the probability that at least one piece of waste in the sample is recyclable is greater than or equal to 0.9999.

A boat rental company for tourism offers its clients two types of boats: sailboat and motorboat.
Furthermore, a client can take the PILOT option. In this case, the boat, whether sailboat or motorboat, is rented with a pilot.
We know that:

• $60\%$ of clients choose a sailboat; among them, $20\%$ take the PILOT option.
• $42\%$ of clients take the PILOT option.

A client is chosen at random and we consider the events:

• $V$: ``the client chooses a sailboat'';
• $L$: ``the client takes the PILOT option''.

Part A

1. Represent the situation with a probability tree that you will complete as you go.
2. Calculate the probability that the client chooses a sailboat and does not take the PILOT option.
3. Prove that the probability that the client chooses a motorboat and takes the PILOT option is equal to 0.30.
4. Deduce $P_{\bar{V}}(L)$, the probability of $L$ given that $V$ is not realized.
5. A client has taken the PILOT option. What is the probability that he chose a sailboat? Round to 0.01.

Part B
When a client does not take the PILOT option, the probability that his boat suffers a breakdown is equal to 0.12. This probability is only 0.005 if the client takes the PILOT option. We consider a client. We denote by $A$ the event: ``his boat suffers a breakdown''.

1. Determine $P(L \cap A)$ and $P(\bar{L} \cap A)$.
2. The company rents 1000 boats. How many breakdowns can it expect?

Part C
We recall that the probability that a given client takes the PILOT option is equal to 0.42. We consider a random sample of 40 clients. We denote by $X$ the random variable counting the number of clients in the sample taking the PILOT option.

1. We admit that the random variable $X$ follows a binomial distribution. Give its parameters without justification.
2. Calculate the probability, rounded to $10^{-3}$, that at least 15 clients take the PILOT option.

A car dealership sells vehicles with electric motors and vehicles with thermal engines. Some customers, before visiting the dealership website, consulted the dealership's digital platform. It was observed that:

• $20\%$ of customers are interested in vehicles with electric motors and $80\%$ prefer to purchase a vehicle with a thermal engine;
• when a customer wishes to buy a vehicle with an electric motor, the probability that the customer consulted the digital platform is 0.5;
• when a customer wishes to buy a vehicle with a thermal engine, the probability that the customer consulted the digital platform is 0.375.

Consider the following events:

• $C$: ``a customer consulted the digital platform'';
• $E$: ``a customer wishes to acquire a vehicle with an electric motor'';
• $T$: ``a customer wishes to acquire a vehicle with a thermal engine''.

Customers make choices independently of one another.

1. a. Calculate the probability that a randomly chosen customer wishes to acquire a vehicle with an electric motor and consulted the digital platform.
   A weighted tree diagram may be used. b. Prove that $P(C) = 0.4$. c. Suppose that a customer consulted the digital platform. Calculate the probability that the customer wishes to buy a vehicle with an electric motor.
2. The dealership welcomes an average of 17 clients daily. Let $X$ be the random variable giving the number of clients wishing to acquire a vehicle with an electric motor. a. Specify the nature and parameters of the probability distribution followed by $X$. b. Calculate the probability that at least three of the clients wish to buy a vehicle with an electric motor during a day. Give the result rounded to $10^{-2}$.

Exercise 1 — 4 points Theme: probability Parts A and B can be treated independently Bicycle users in a city are classified into two disjoint categories:

• those who use bicycles for professional travel;
• those who use bicycles only for leisure.

A survey gives the following results:

• $21\%$ of users are under 35 years old. Among them, $68\%$ use their bicycle only for leisure while the others use it for professional travel;
• among those 35 years or older, only $20\%$ use their bicycle for professional travel, the others use it only for leisure.

A bicycle user from this city is randomly interviewed. Throughout the exercise, the following events are considered:

• $J$: ``the person interviewed is under 35 years old'';
• $T$: ``the person interviewed uses the bicycle for professional travel'';
• $\bar{J}$ and $\bar{T}$ are the complementary events of $J$ and $T$.

Part A

1. Calculate the probability that the person interviewed is under 35 years old and uses their bicycle for professional travel. You may use a probability tree.
2. Calculate the exact value of the probability of $T$.
3. Now consider a resident who uses their bicycle for professional travel. Prove that the probability that they are under 35 years old is 0.30 to within $10^{-2}$.

Part B In this part, we are interested only in people using their bicycle for professional travel. We assume that $30\%$ of them are under 35 years old.
A sample of 120 people is randomly selected from among them to complete an additional questionnaire. The selection of this sample is treated as random sampling with replacement. Each individual in this sample is asked their age. $X$ represents the number of people in the sample who are under 35 years old. In this part, results should be rounded to $10^{-3}$.

1. Determine the nature and parameters of the probability distribution followed by $X$.
2. Calculate the probability that at least 50 bicycle users among the 120 are under 35 years old.

A car dealership sells two types of vehicles:

• $60\%$ are fully electric vehicles;
• $40\%$ are rechargeable hybrid vehicles.

$75\%$ of buyers of fully electric vehicles and $52\%$ of buyers of rechargeable hybrid vehicles have the material possibility of installing a charging station at home.
A buyer is chosen at random and the following events are considered:

• $E$: ``the buyer chooses a fully electric vehicle'';
• $B$: ``the buyer has the possibility of installing a charging station at home''.

Throughout the exercise, probabilities should be rounded to the nearest thousandth if necessary.

1. Calculate the probability that the buyer chooses a fully electric vehicle and has the possibility of installing a charging station at home.
   A weighted tree diagram may be used.
2. Prove that $P(B) = 0.658$.
3. A buyer has the possibility of installing a charging station at home. What is the probability that he chooses a fully electric vehicle?
4. A sample of 20 buyers is chosen. This sampling is treated as drawing with replacement. Let $X$ be the random variable that gives the total number of buyers able to install a charging station at home among the sample of 20 buyers. a. Determine the nature and parameters of the probability distribution followed by $X$. b. Calculate $P(X = 8)$. c. Calculate the probability that at least 10 buyers can install a charging station. d. Calculate the expected value of $X$. e. The dealership manager decides to offer the installation of the charging station to buyers who have the possibility of installing one at home. This installation costs $1200$~\euro. On average, what amount should she plan to spend on this offer when selling 20 vehicles?

A survey conducted in France provides the following information:

• $60\%$ of people over 15 years old intend to watch the Paris 2024 Olympic and Paralympic Games (OPG) on television;
• among those who intend to watch the OPG, 8 out of 9 people declare that they regularly practice a sport.

A person over 15 years old is chosen at random. The following events are considered:

• $J$: ``the person intends to watch the Paris 2024 OPG on television'';
• $S$: ``the chosen person declares that they regularly practice a sport''.

We denote by $\bar{J}$ and $\bar{S}$ their complementary events.
In questions 1. and 2., probabilities will be given in the form of an irreducible fraction.

1. Demonstrate that the probability that the chosen person intends to watch the Paris 2024 OPG on television and declares that they regularly practice a sport is $\frac{8}{15}$. A weighted tree diagram may be used.
   According to this survey, two out of three people over 15 years old declare that they regularly practice a sport.
   
2. [2.] a. Calculate the probability that the chosen person does not intend to watch the Paris 2024 OPG on television and declares that they regularly practice a sport. b. Deduce the probability of $S$ given $\bar{J}$ denoted $P_{\bar{J}}(S)$.
   In the rest of the exercise, results will be rounded to the nearest thousandth.
   
3. [3.] As part of a promotional operation, 30 people over 15 years old are chosen at random. This choice is treated as sampling with replacement. Let $X$ be the random variable that gives the number of people declaring that they regularly practice a sport among the 30 people. a. Determine the nature and parameters of the probability distribution followed by $X$. b. Calculate the probability that exactly 16 people declare that they regularly practice a sport among the 30 people. c. The French judo federation wishes to offer a ticket for the final of the mixed team judo event at the Arena Champ-de-Mars for each person declaring that they regularly practice a sport among these 30 people. The price of a ticket is $380\,€$ and a budget of 10000 euros is available for this operation. What is the probability that this budget is insufficient?

In tennis, the player who is serving can, in case of failure on the first serve, serve a second ball. In match play, Abel succeeds with his first serve in $70\%$ of cases. When the first serve is successful, he wins the point in $80\%$ of cases. On the other hand, after a failure on his first serve, Abel wins the point in $45\%$ of cases. Abel is serving. Consider the following events:

• S: ``Abel succeeds with his first serve''
• G: ``Abel wins the point''.

1. Describe the event $S$ then translate the situation with a probability tree.
2. Calculate $P(S \cap G)$.
3. Justify that the probability of event $G$ is equal to 0.695.
4. Abel has won the point. What is the probability that he succeeded with his first serve?
5. Are events $S$ and $G$ independent? Justify.