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.

Exercise 1 Probability
The alarm system of a company operates in such a way that, if a danger presents itself, the alarm activates with a probability of 0.97. The probability that a danger presents itself is 0.01 and the probability that the alarm activates is 0.01465. We denote $A$ the event ``the alarm activates'' and $D$ the event ``a danger presents itself''. We denote $\bar{M}$ the opposite event of an event $M$ and $P(M)$ the probability of the event $M$.

PART A

1. Represent the situation with a weighted tree diagram that will be completed as the exercise progresses.
2. a. Calculate the probability that a danger presents itself and the alarm activates. b. Deduce from this the probability that a danger presents itself given that the alarm activates. Round the result to $10^{-3}$.
3. Show that the probability that the alarm activates given that no danger has presented itself is 0.005.
4. An alarm is considered not to function normally when a danger presents itself and it does not activate, or when no danger presents itself and it activates. Show that the probability that the alarm does not function normally is less than 0.01.

PART B

A factory manufactures alarm systems in large quantities. We successively and randomly select 5 alarm systems from the factory's production. This selection is treated as sampling with replacement. We denote $S$ the event ``the alarm does not function normally'' and we admit that $P(S) = 0.00525$. We consider $X$ the random variable that gives the number of alarm systems not functioning normally among the 5 alarm systems selected. Results should be rounded to $10^{-4}$.

1. Give the probability distribution followed by the random variable $X$ and specify its parameters.
2. Calculate the probability that, in the selected batch, only one alarm system does not function normally.
3. Calculate the probability that, in the selected batch, at least one alarm system does not function normally.

PART C

Let $n$ be a non-zero natural integer. We successively and randomly select $n$ alarm systems. This selection is treated as sampling with replacement. Determine the smallest integer $n$ such that the probability of having, in the selected batch, at least one alarm system that does not function normally is greater than 0.07.

Exercise 1: Probability
A company manufactures components for the automotive industry. These components are designed on three assembly lines numbered 1 to 3.

• Half of the components are designed on line $\mathrm{n}^{\circ}1$;
• $30\%$ of the components are designed on line $\mathrm{n}^{\circ}2$;
• the remaining components are designed on line $\mathrm{n}^{\circ}3$.

At the end of the manufacturing process, it appears that $1\%$ of the parts from line $\mathrm{n}^{\circ}1$ have a defect, as do $0.5\%$ of the parts from line $\mathrm{n}^{\circ}2$ and $4\%$ of the parts from line $\mathrm{n}^{\circ}3$. One of these components is randomly selected. We denote:

• $C_1$ the event ``the component comes from line $\mathrm{n}^{\circ}1$'';
• $C_2$ the event ``the component comes from line $\mathrm{n}^{\circ}2$'';
• $C_3$ the event ``the component comes from line $\mathrm{n}^{\circ}3$'';
• $D$ the event ``the component is defective'' and $\bar{D}$ its complementary event.

Throughout the exercise, probability calculations will be given as exact decimal values or rounded to $10^{-4}$ if necessary.
PART A

1. Represent this situation with a probability tree.
2. Calculate the probability that the selected component comes from line $\mathrm{n}^{\circ}3$ and is defective.
3. Show that the probability of event $D$ is $P(D) = 0.0145$.
4. Calculate the probability that a defective component comes from line $\mathrm{n}^{\circ}3$.

PART B
The company decides to package the produced components by forming batches of $n$ units. We denote $X$ the random variable which, to each batch of $n$ units, associates the number of defective components in this batch. Given the company's production and packaging methods, we can consider that $X$ follows the binomial distribution with parameters $n$ and $p = 0.0145$.

1. In this question, the batches contain 20 units. We set $n = 20$. a. Calculate the probability that a batch contains exactly three defective components. b. Calculate the probability that a batch contains no defective components. Deduce the probability that a batch contains at least one defective component.
2. The company director wishes the probability of having no defective components in a batch of $n$ components to be greater than 0.85. He proposes to form batches of at most 11 components. Is he correct? Justify your answer.

PART C
The manufacturing costs of the components of this company are 15 euros if they come from assembly line $\mathrm{n}^{\circ}1$, 12 euros if they come from assembly line $\mathrm{n}^{\circ}2$ and 9 euros if they come from assembly line $\mathrm{n}^{\circ}3$. Calculate the average manufacturing cost of a component for this company.

During a fair, a game organizer has, on one hand, a wheel with four white squares and eight red squares and, on the other hand, a bag containing five tokens bearing the numbers $1, 2, 3, 4$ and 5. The game consists of spinning the wheel, each square having equal probability of being obtained, then extracting one or two tokens from the bag according to the following rule:

• if the square obtained by the wheel is white, then the player extracts one token from the bag;
• if the square obtained by the wheel is red, then the player extracts successively and without replacement two tokens from the bag.

The player wins if the token(s) drawn all bear an odd number.

1. A player plays one game and we denote by $B$ the event ``the square obtained is white'', $R$ the event ``the square obtained is red'' and $G$ the event ``the player wins the game''. a. Give the value of the conditional probability $P _ { B } ( G )$. b. It is admitted that the probability of drawing successively and without replacement two odd tokens is equal to 0.3. Copy and complete the following probability tree.
2. a. Show that $P ( G ) = 0.4$. b. A player wins the game. What is the probability that he obtained a white square by spinning the wheel?
3. Are the events $B$ and $G$ independent? Justify.
4. The same player plays ten games. The tokens drawn are returned to the bag after each game. We denote by $X$ the random variable equal to the number of games won. a. Explain why $X$ follows a binomial distribution and specify its parameters. b. Calculate the probability, rounded to $10 ^ { - 3 }$, that the player wins exactly three games out of the ten games played. c. Calculate $P ( X \geqslant 4 )$ rounded to $10 ^ { - 3 }$. Give an interpretation of the result obtained.
5. A player plays $n$ games and we denote by $p _ { n }$ the probability of the event ``the player wins at least one game''. a. Show that $p _ { n } = 1 - 0.6 ^ { n }$. b. Determine the smallest value of the integer $n$ for which the probability of winning at least one game is greater than or equal to 0.99.

Exercise 1 (7 points) — Main topics covered: Probability
In basketball, there are two types of shots:

• two-point shots: taken near the basket and score two points if successful.
• three-point shots: taken far from the basket and score three points if successful.

Stéphanie is practising shooting. We have the following data:

• One quarter of her shots are two-point shots. Among these, $60\%$ are successful.
• Three quarters of her shots are three-point shots. Among these, $35\%$ are successful.

1. Stéphanie takes a shot. Consider the following events: $D$: ``It is a two-point shot''. $R$: ``the shot is successful''. a. Represent the situation using a probability tree. b. Calculate the probability $p(\bar{D} \cap R)$. c. Prove that the probability that Stéphanie successfully makes a shot is equal to 0.4125. d. Stéphanie successfully makes a shot. Calculate the probability that it is a three-point shot. Round the result to the nearest hundredth.
   
2. Stéphanie now takes a series of 10 three-point shots. Let $X$ be the random variable that counts the number of successful shots. Consider that the shots are independent. Recall that the probability that Stéphanie successfully makes a three-point shot is equal to 0.35. a. Justify that $X$ follows a binomial distribution. Specify its parameters. b. Calculate the expected value of $X$. Interpret the result in the context of the exercise. c. Determine the probability that Stéphanie misses 4 or more shots. Round the result to the nearest hundredth. d. Determine the probability that Stéphanie misses at most 4 shots. Round the result to the nearest hundredth.
   
3. Let $n$ be a non-zero natural number. Stéphanie wishes to take a series of $n$ three-point shots. Consider that the shots are independent. Recall that the probability that she successfully makes a three-point shot is equal to 0.35. Determine the minimum value of $n$ so that the probability that Stéphanie successfully makes at least one shot among the $n$ shots is greater than or equal to 0.99.

Exercise 1 — 6 points

Main topics covered: Probability

At a ski resort, there are two types of passes depending on the skier's age:

• a JUNIOR pass for people under 25 years old;
• a SENIOR pass for others.

Furthermore, a user can choose, in addition to the pass corresponding to their age, the skip-the-line option which allows them to reduce waiting time at the ski lifts. We assume that:

• $20 \%$ of skiers have a JUNIOR pass;
• $80 \%$ of skiers have a SENIOR pass;
• among skiers with a JUNIOR pass, $6 \%$ choose the skip-the-line option;
• among skiers with a SENIOR pass, $12.5 \%$ choose the skip-the-line option.

We interview a skier at random and consider the events:

• $J$ : ``the skier has a JUNIOR pass'';
• $C$ : ``the skier chooses the skip-the-line option''.

The two parts can be worked on independently

Part A

1. Represent the situation with a probability tree.
2. Calculate the probability $P ( J \cap C )$.
3. Prove that the probability that the skier chooses the skip-the-line option is equal to 0.112.
4. The skier has chosen the skip-the-line option. What is the probability that this is a skier with a SENIOR pass? Round the result to $10 ^ { - 3 }$.
5. Is it true that people under twenty-five years old represent less than $15 \%$ of skiers who chose the skip-the-line option? Explain.

Part B

We recall that the probability that a skier chooses the skip-the-line option is equal to 0.112. We consider a sample of 30 skiers chosen at random. Let $X$ be the random variable that counts the number of skiers in the sample who chose the skip-the-line option.

1. We assume that the random variable $X$ follows a binomial distribution. Give the parameters of this distribution.
2. Calculate the probability that at least one of the 30 skiers chose the skip-the-line option. Round the result to $10 ^ { - 3 }$.
3. Calculate the probability that at most one of the 30 skiers chose the skip-the-line option. Round the result to $10 ^ { - 3 }$.
4. Calculate the expected value of the random variable $X$.

Exercise 1 — 7 points

Topics: Probability

In Hugo's shop, customers can rent two types of bicycles: road bikes or mountain bikes. Each type of bicycle can be rented in an electric version or not.
A customer is chosen at random from the shop, and we assume that:

• If the customer rents a road bike, the probability that it is an electric bike is 0.4;
• If the customer rents a mountain bike, the probability that it is an electric bike is 0.7;
• The probability that the customer rents an electric bike is 0.58.

We denote by $\alpha$ the probability that the customer rents a road bike, with $0 \leqslant \alpha \leqslant 1$. We consider the following events:

• R: ``the customer rents a road bike'';
• $E$ : ``the customer rents an electric bike'';
• $\bar { R }$ and $\bar { E }$, complementary events of $R$ and $E$.

We model this random situation using the tree shown below. If $F$ denotes any event, we denote by $p ( F )$ the probability of $F$.

1. Copy this tree onto your answer sheet and complete it.
2. a. Show that $p ( E ) = 0.7 - 0.3 \alpha$. b. Deduce that: $\alpha = 0.4$.
3. We know that the customer rented an electric bike. Determine the probability that they rented a mountain bike. Give the result rounded to the nearest hundredth.
4. What is the probability that the customer rents an electric mountain bike?
5. The daily rental price of a non-electric road bike is 25 euros, that of a non-electric mountain bike is 35 euros. For each type of bike, choosing the electric version increases the daily rental price by 15 euros. We denote by $X$ the random variable modeling the daily rental price of a bike. a. Give the probability distribution of $X$. Present the results in the form of a table. b. Calculate the expected value of $X$ and interpret this result.
6. When 30 of Hugo's customers are chosen at random, we treat this choice as sampling with replacement. We denote by $Y$ the random variable associating to a sample of 30 randomly chosen customers the number of customers who rent an electric bike. We recall that the probability of event $E$ is: $p ( E ) = 0.58$. a. Justify that $Y$ follows a binomial distribution and specify its parameters. b. Determine the probability that a sample contains exactly 20 customers who rent an electric bike. Give the result rounded to the nearest thousandth. c. Determine the probability that a sample contains at least 15 customers who rent an electric bike. Give the result rounded to the nearest thousandth.

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?

Exercise 4 — 7 points
Theme: Probability
During the manufacture of a pair of glasses, the pair of lenses must undergo two treatments denoted T1 and T2.
Part A
A pair of lenses is randomly selected from production. We denote by $A$ the event: ``the pair of lenses has a defect for treatment T1''. We denote by $B$ the event: ``the pair of lenses has a defect for treatment T2''. We denote by $\bar{A}$ and $\bar{B}$ respectively the complementary events of $A$ and $B$.
A study has shown that:

• the probability that a pair of lenses has a defect for treatment T1, denoted $P(A)$, is equal to 0.1.
• the probability that a pair of lenses has a defect for treatment T2, denoted $P(B)$, is equal to 0.2.
• the probability that a pair of lenses has neither of the two defects is 0.75.

1. Copy and complete the following table with the corresponding probabilities.
   

   	$A$	$\bar{A}$	Total
   $B$
   $\bar{B}$
   Total			1

   
   
2. a. Determine, by justifying the answer, the probability that a pair of lenses, randomly selected from production, has a defect for at least one of the two treatments T1 or T2. b. Give the probability that a pair of lenses, randomly selected from production, has two defects, one for each treatment T1 and T2. c. Are the events $A$ and $B$ independent? Justify the answer.
   
3. Calculate the probability that a pair of lenses, randomly selected from production, has a defect for only one of the two treatments.
   
4. Calculate the probability that a pair of lenses, randomly selected from production, has a defect for treatment T2, given that this pair of lenses has a defect for treatment T1.

Part B
A sample of 50 pairs of lenses is randomly selected from production. We assume that the production is large enough to assimilate this selection to a draw with replacement. We denote by $X$ the random variable which, to each sample of this type, associates the number of pairs of lenses that have the defect for treatment T1.

1. Justify that the random variable $X$ follows a binomial distribution and specify the parameters of this distribution.
   
2. Give the expression allowing the calculation of the probability of having, in such a sample, exactly 10 pairs of lenses that have this defect. Perform this calculation and round the result to $10^{-3}$.
   
3. On average, how many pairs of lenses with this defect can be found in a sample of 50 pairs?

4 points Parts $\boldsymbol{A}$ and $\boldsymbol{B}$ are independent. The requested probabilities will be given to $10^{-3}$ near. To help detect certain allergies, a blood test can be performed whose result is either positive or negative. In a population, this test gives the following results:

• If an individual is allergic, the test is positive in $97\%$ of cases;
• If an individual is not allergic, the test is negative in $95{,}7\%$ of cases.

Furthermore, $20\%$ of individuals in the concerned population have a positive test. We randomly choose an individual from the population, and we denote:

• $A$ the event ``the individual is allergic'';
• $T$ the event ``the individual has a positive test''.

We denote by $\bar{A}$ and $\bar{T}$ the complementary events of $A$ and $T$. We also call $x$ the probability of event $A$: $x = p(A)$.
Part A

1. Reproduce and complete the tree describing the situation, indicating on each branch the corresponding probability.
2. a. Prove the equality: $p(T) = 0{,}927x + 0{,}043$. b. Deduce the probability that the chosen individual is allergic.
3. Justify by a calculation the following statement: ``If the test of an individual chosen at random is positive, there is more than $80\%$ chance that this individual is allergic''.

Part B
A survey on allergies is conducted in a city by interviewing 150 randomly chosen residents, and we assume that this choice amounts to successive independent draws with replacement. We know that the probability that a randomly chosen resident in this city is allergic is equal to $0{,}08$. We denote by $X$ the random variable that associates to a sample of 150 randomly chosen residents the number of allergic people in this sample.

1. What is the probability distribution followed by the random variable $X$? Specify its parameters.
2. Determine the probability that exactly 20 people among the 150 interviewed are allergic.
3. Determine the probability that at least $10\%$ of the people among the 150 interviewed are allergic.

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.

Paratuberculosis is an infectious digestive disease that affects cows. It is caused by the presence of a bacterium in the cow's intestine.
A study is conducted in a region where $0.4\%$ of the cow population is infected.
There is a test that reveals the immune response of an organism infected by the bacterium. The result of this test can be either ``positive'' or ``negative''.
A cow is chosen at random from the region. Given the characteristics of the test, we know that:

• If the cow is affected by the infection, the probability that its test is positive is 0.992;
• If the cow is not affected by the infection, the probability that its test is negative is 0.984.

We denote by $I$ the event ``the cow is affected by the infection'' and $T$ the event ``the cow presents a positive test''. We denote by $\bar{I}$ the complementary event of $I$ and $\bar{T}$ the complementary event of $T$.
Part A

1. Reproduce and complete the weighted tree below modelling the situation.
2. a. Calculate the probability that the cow is not affected by the infection and that its test is negative. Give the result to $10^{-3}$ near. b. Show that the probability, to $10^{-3}$ near, that the cow presents a positive test is approximately equal to 0.020. c. The ``positive predictive value of the test'' is the probability that the cow is affected by the infection given that its test is positive. Calculate the positive predictive value of this test. Give the result to $10^{-3}$ near. d. The test gives incorrect information about the cow's state of health when the cow is not infected and presents a positive test result or when the cow is infected and presents a negative test result. Calculate the probability that this test gives incorrect information about the cow's state of health. Give a result to $10^{-3}$ near.

Part B

1. When a sample of 100 cows is chosen at random from the region, this choice is treated as a draw with replacement. Recall that, for a cow chosen at random from the region, the probability that the test is positive is equal to 0.02. We denote by $X$ the random variable that associates to a sample of 100 cows from the region chosen at random the number of cows presenting a positive test in this sample. a. What is the probability distribution followed by the random variable $X$? Justify the answer and specify the parameters of this distribution. b. Calculate the probability that in a sample of 100 cows, there are exactly 3 cows presenting a positive test. Give a result to $10^{-3}$ near. c. Calculate the probability that in a sample of 100 cows, there are at most 3 cows presenting a positive test. Give a result to $10^{-3}$ near.
2. We now choose a sample of $n$ cows from this region, $n$ being a non-zero natural integer. We admit that this choice can be treated as a draw with replacement. Determine the minimum value of $n$ so that the probability that there is, in the sample, at least one cow tested positive, is greater than or equal to 0.99.

A technician controls the machines equipping a large company. All these machines are identical. We know that:

• $20\%$ of machines are under warranty;
• $0.2\%$ of machines are both defective and under warranty;
• $8.2\%$ of machines are defective.

The technician tests a machine at random. We consider the following events:

• G: ``the machine is under warranty'';
• $D$: ``the machine is defective'';
• $\bar{G}$ and $\bar{D}$ denote respectively the complementary events of $G$ and $D$.

The probability $p_{G}(D)$ of event $D$ given that $G$ is realized is equal to: a. 0.002 b. 0.01 c. 0.024 d. 0.2

A video game has a large community of online players. Before starting a game, the player must choose between two ``worlds'': either world A or world B. An individual is chosen at random from the community of players. When playing a game, we assume that:

• the probability that the player chooses world A is equal to $\frac{2}{5}$;
• if the player chooses world A, the probability that they win the game is $\frac{7}{10}$;
• the probability that the player wins the game is $\frac{12}{25}$.

We consider the following events:

• A: ``The player chooses world A'';
• B: ``The player chooses world B'';
• G: ``The player wins the game''.

This exercise is a multiple choice questionnaire (5 questions). For each question, only one of the four proposed answers is correct.

1. The probability that the player chooses world A and wins the game is equal to: a. $\frac{7}{10}$ b. $\frac{3}{25}$ c. $\frac{7}{25}$ d. $\frac{24}{125}$
2. The probability $P_{B}(G)$ of event $G$ given that $B$ is realized is equal to: a. $\frac{1}{5}$ b. $\frac{1}{3}$ c. $\frac{7}{15}$ d. $\frac{5}{12}$

In the rest of the exercise, a player plays 10 successive games. This situation is treated as a random draw with replacement. We recall that the probability of winning a game is $\frac{12}{25}$.
3. The probability, rounded to the nearest thousandth, that the player wins exactly 6 games is equal to: a. 0.859 b. 0.671 c. 0.188 d. 0.187
4. We consider a natural number $n$ for which the probability, rounded to the nearest thousandth, that the player wins at most $n$ games is 0.207. Then: a. $n = 2$ b. $n = 3$ c. $n = 4$ d. $n = 5$
5. The probability that the player wins at least one game is equal to: a. $1 - \left(\frac{12}{25}\right)^{10}$ b. $\left(\frac{13}{25}\right)^{10}$ c. $\left(\frac{12}{25}\right)^{10}$ d. $1 - \left(\frac{13}{25}\right)^{10}$

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 company calls people by telephone to sell them a product.

• The company calls each person a first time:
• the probability that the person does not answer is equal to 0.6;
• if the person answers, the probability that they buy the product is equal to 0.3.
• If the person did not answer on the first call, a second call is made:
• the probability that the person does not answer is equal to 0.3;
• if the person answers, the probability that they buy the product is equal to 0.2.
• If a person does not answer on the second call, we stop contacting them.

We choose a person at random and consider the following events: $D _ { 1 }$: ``the person answers on the first call''; $D _ { 2 }$: ``the person answers on the second call''; $A$: ``the person buys the product''.

Part A

1. Copy and complete the weighted tree opposite.
2. Using the weighted tree, show that the probability of event $A$ is $P ( A ) = 0.204$.
3. We know that the person bought the product. What is the probability that they answered on the first call?

Part B

We recall that, for a given person, the probability that they buy the product is equal to 0.204.

1. We consider a random sample of 30 people. Let $X$ be the random variable that gives the number of people in the sample who buy the product. a. We admit that $X$ follows a binomial distribution. Give, without justification, its parameters. b. Determine the probability that exactly 6 people in the sample buy the product. Round the result to the nearest thousandth. c. Calculate the expected value of the random variable $X$. Interpret the result.
2. Let $n$ be a non-zero natural number. We now consider a sample of $n$ people. Determine the smallest value of $n$ such that the probability that at least one person in the sample buys the product is greater than or equal to 0.99.

A technician controls the machines equipping a large company. All these machines are identical. We know that:

• $20\%$ of machines are under warranty;
• $0.2\%$ of machines are both defective and under warranty;
• $8.2\%$ of machines are defective.

The technician tests a machine at random. We consider the following events:

• G: ``the machine is under warranty'';
• $D$: ``the machine is defective'';
• $\bar{G}$ and $\bar{D}$ denote respectively the complementary events of $G$ and $D$.

The probability $p(\bar{G} \cap D)$ is equal to: a. 0.01 b. 0.08 c. 0.1 d. 0.21

A technician controls the machines equipping a large company. All these machines are identical. We know that:

• $20\%$ of machines are under warranty;
• $0.2\%$ of machines are both defective and under warranty;
• $8.2\%$ of machines are defective.

The technician tests a machine at random. We consider the following events:

• G: ``the machine is under warranty'';
• $D$: ``the machine is defective'';
• $\bar{G}$ and $\bar{D}$ denote respectively the complementary events of $G$ and $D$.

The machine is defective. The probability that it is under warranty is approximately equal, to $10^{-3}$ near, to: a. 0.01 b. 0.024 c. 0.082 d. 0.1

For each of the five questions in this exercise, only one of the four proposed answers is correct. The candidate will indicate on his/her work the number of the question and the chosen answer. No justification is required.
A wrong answer, multiple answers, or the absence of an answer to a question neither gives nor removes points.
An urn contains 15 indistinguishable balls to the touch, numbered from 1 to 15. The ball numbered 1 is red. The balls numbered 2 to 5 are blue. The other balls are green. We choose a ball at random from the urn. We denote $R$ (respectively $B$ and $V$) the event: ``The ball drawn is red'' (respectively blue and green).
Question 1: What is the probability that the ball drawn is blue or numbered with an even number?

Answer A	Answer B	Answer C	Answer D
$\frac { 7 } { 15 }$	$\frac { 9 } { 15 }$	$\frac { 11 } { 10 }$	None of the previous statements is correct.

Question 2: Given that the ball drawn is green, what is the probability that it is numbered 7?

Answer A	Answer B	Answer C	Answer D
$\frac { 1 } { 15 }$	$\frac { 7 } { 15 }$	$\frac { 1 } { 10 }$	None of the previous statements is correct.

A game is set up. To be able to play, the player pays the sum of 10 euros called the stake. This game consists of drawing a ball at random from the urn.

• If the ball drawn is blue, the player wins, in euros, three times the number of the ball.
• If the ball drawn is green, the player wins, in euros, the number of the ball.
• If the ball drawn is red, the player wins nothing.

We denote $G$ the random variable that gives the algebraic gain of the player, that is, the difference between what he wins and his initial stake. For example, if the player draws the blue ball numbered 3, then his algebraic gain is $-1$ euro.
Question 3: What is the value of $P ( G = 5 )$ ?

Answer A	Answer B	Answer C	Answer D
$\frac { 1 } { 15 }$	$\frac { 2 } { 15 }$	$\frac { 1 } { 3 }$	None of the previous statements is correct.

Question 4: What is the value of $P _ { R } ( G = 0 )$ ?

Answer A	Answer B	Answer C	Answer D
0	$\frac { 1 } { 15 }$	1	None of the previous statements is correct.

Question 5: What is the value of $P _ { ( G = - 4 ) } ( V )$ ?

Answer A	Answer B	Answer C	Answer D
$\frac { 1 } { 15 }$	$\frac { 4 } { 15 }$	$\frac { 1 } { 2 }$	None of the previous statements is correct.

This exercise is a multiple choice questionnaire. For each of the following questions, only one of the four proposed answers is correct. To answer, indicate on your paper the number of the question and the letter of the chosen answer. No justification is required.
A wrong answer, an absence of answer, or a multiple answer, neither gives nor removes points.
The 200 members of a club are girls or boys. These members practice rowing or basketball according to the distribution shown in the table below.

\cline { 2 - 4 } \multicolumn{1}{c|}{}	Rowing	Basketball	Total
Girls	25	80	105
Boys	50	45	95
Total	75	125	200

We choose a member at random and consider the following events: $F$ : the member is a girl; $A$ : the member practices rowing.

1. The probability of $F$ given $A$ is equal to : a. $\frac { 25 } { 100 }$ b. $\frac { 25 } { 75 }$ c. $\frac { 25 } { 105 }$ d. $\frac { 75 } { 105 }$
2. The probability of the event $A \cup F$ is equal to : a. $\frac { 9 } { 10 }$ b. $\frac { 1 } { 8 }$ c. $\frac { 31 } { 40 }$ d. $\frac { 5 } { 36 }$

To get to work, Albert can use either the bus or the train. The probability that the bus breaks down is equal to $b$. The probability that the train breaks down is equal to $t$. Bus and train breakdowns occur independently.
3. The probability $p _ { 1 }$ that the bus or the train breaks down is equal to : a. $p _ { 1 } = b t$ b. $p _ { 1 } = 1 - b t$ c. $p _ { 1 } = b + t$ d. $p _ { 1 } = b + t - b t$
4. The probability $p _ { 2 }$ that Albert can get to work is equal to : a. $p _ { 2 } = b t$ b. $p _ { 2 } = 1 - b t$ c. $p _ { 2 } = b + t$ d. $p _ { 2 } = b + t - b t$
5. We consider a coin for which the probability of obtaining HEADS is equal to $x$. We flip the coin $n$ times. The flips are independent. The probability $p$ of obtaining at least one HEADS in the $n$ flips is equal to a. $p = x ^ { n }$ b. $p = ( 1 - x ) ^ { n }$ c. $p = 1 - x ^ { n }$ d. $p = 1 - ( 1 - x ) ^ { n }$

Here is the distribution of the main blood groups of the inhabitants of France. $A+, O+, B+, A-, O-, AB+, B-$ and $AB-$ are the different blood groups combined with the Rh factor.
A random experiment consists of choosing a person at random from the French population and determining their blood group and Rh factor.
We adopt notations of the type: $A+$ is the event ``the person has blood group A and Rh factor $+$'' $A-$ is the event ``the person has blood group A and Rh factor $-$'' $A$ is the event ``the person has blood group A''
Parts 1 and 2 are independent.
Part 1
We denote $Rh+$ the event ``The person has positive Rh factor''.

1. Justify that the probability that the chosen person has positive Rh factor is equal to 0.849.
2. Demonstrate using the data from the problem statement that $P_{Rh_+}(A) = 0.450$ to 0.001 near.
3. A person remembers that their blood group is AB but has forgotten their Rh factor. What is the probability that their Rh factor is negative? Round the result to 0.001 near.

Part 2
In this part, results will be rounded to 0.001 near.
A universal blood donor is a person with blood group O and negative Rh factor. Recall that $6.5\%$ of the French population has blood group $O-$.

1. We consider 50 people chosen at random from the French population and we denote $X$ the random variable that counts the number of universal donors. a. Determine the probability that 8 people are universal donors. Justify your answer. b. Consider the function below named \texttt{proba} with argument \texttt{k} written in Python language. \begin{verbatim} def proba(k) : p = 0 for i in range(k+1) : p = p + binomiale(i,50,0.065) return p \end{verbatim} This function uses the binomial function with arguments $i, n$ and $p$, created for this purpose, which returns the value of the probability $P(X=i)$ in the case where $X$ follows a binomial distribution with parameters $n$ and $p$. Determine the numerical value returned by the \texttt{proba} function when you enter \texttt{proba(8)} in the Python console. Interpret this result in the context of the exercise.
2. What is the minimum number of people to choose at random from the French population so that the probability that at least one of the chosen people is a universal donor is greater than 0.999.

In the journal Lancet Public Health, researchers claim that on May 11, 2020, 5.7\% of French adults had already been infected with COVID 19.

1. An individual is drawn from the adult French population on May 11, 2020. Let $I$ be the event: ``the adult has already been infected with COVID 19''. What is the probability that this individual drawn has already been infected with COVID 19?
2. A sample of 100 people from the population is drawn, assumed to be chosen independently of each other. This sampling is assimilated to a draw with replacement. Let $X$ be the random variable that counts the number of people who have already been infected. a. Justify that $X$ follows a binomial distribution and give its parameters. b. Calculate its mathematical expectation. Interpret this result in the context of the exercise. c. What is the probability that there is no infected person in the sample? Give an approximate value to $10^{-4}$ near of the result. d. What is the probability that there are at least 2 infected people in the sample? Give an approximate value to $10^{-4}$ near of the result. e. Determine the smallest integer $n$ such that $P(X \leq n) > 0.9$. Interpret this result in the context of the exercise.

To access a company's private network from outside, employee connections are randomly routed through three different remote servers, denoted $\mathrm{A}, \mathrm{B}$ and C. These servers have different technical characteristics and connections are distributed as follows:

• $25\%$ of connections are routed through server A;
• $15\%$ of connections are routed through server B;
• the remaining connections are made through server C.

A connection is said to be stable if the user does not experience a disconnection after authentication to the servers. The IT maintenance team has statistically observed that, under normal server operation:

• $90\%$ of connections via server A are stable;
• $80\%$ of connections via server B are stable;
• $85\%$ of connections via server C are stable.

Part A
We are interested in the state of a connection made by an employee of the company. We consider the following events:

• A: ``The connection was made via server A'';
• B: ``The connection was made via server B'';
• C: ``The connection was made via server C'';
• S: ``The connection is stable''.

We denote by $\bar{S}$ the complementary event of event $S$.

1. Copy and complete the weighted tree below modelling the situation described in the problem.
2. Prove that the probability that the connection is stable and passes through server B is equal to 0.12.
3. Calculate the probability $P(C \cap \bar{S})$ and interpret the result in the context of the exercise.
4. Prove that the probability of event $S$ is $P(S) = 0.855$.
5. Now suppose that the connection is stable. Calculate the probability that the connection was made from server B. Give the answer rounded to the nearest thousandth.

Part B
According to Part A, the probability that a connection is unstable is equal to 0.145.

1. In order to detect server malfunctions, we study a sample of 50 connections to the network, these connections being chosen at random. We assume that the number of connections is large enough that this choice can be treated as sampling with replacement.
   Let $X$ denote the random variable equal to the number of unstable connections to the company's network, in this sample of 50 connections. a. We admit that the random variable $X$ follows a binomial distribution. Specify its parameters. b. Give the probability that at most eight connections are unstable. Give the answer rounded to the nearest thousandth.
2. In this question, we now form a sample of $n$ connections, still under the same conditions, where $n$ denotes a strictly positive natural number. We denote by $X_n$ the random variable equal to the number of unstable connections and we admit that $X_n$ follows a binomial distribution with parameters $n$ and 0.145. a. Give the expression as a function of $n$ of the probability $p_n$ that at least one connection in this sample is unstable. b. Determine, by justifying, the smallest value of the natural number $n$ such that the probability $p_n$ is greater than or equal to 0.99.
3. We are interested in the random variable $F_n$ equal to the frequency of unstable connections in a sample of $n$ connections, where $n$ denotes a strictly positive natural number. We thus have $F_n = \frac{X_n}{n}$, where $X_n$ is the random variable defined in question 2. a. Calculate the expectation $E(F_n)$. We admit that $V(F_n) = \frac{0.123975}{n}$. b. Verify that: $P\left(\left|F_n - 0.145\right| \geqslant 0.1\right) \leqslant \frac{12.5}{n}$ c. A company manager studies a sample of 1000 connections and observes that for this sample $F_{1000} = 0.3$. He suspects a server malfunction. Is he right?

Exercise 1 — Part A
The centre offers people coming for a weekend an introductory roller skating formula consisting of two training sessions. We randomly choose a person among those who have subscribed to this formula. We denote by $A$ and $B$ the following events:

• A: ``The person falls during the first session'';
• B: ``The person falls during the second session''.

For any event $E$, we denote $P(E)$ its probability and $\bar{E}$ its complementary event. Observations allow us to assume that $P(A) = 0{,}6$. Furthermore, we observe that:

• If the person falls during the first session, the probability that they fall during the second is 0.3;
• If the person does not fall during the first session, the probability that they fall during the second is 0.4.

1. Represent the situation with a probability tree.
2. Calculate the probability $P(\bar{A} \cap \bar{B})$ and interpret the result.
3. Show that $P(B) = 0{,}34$.
4. The person does not fall during the second training session. Calculate the probability that they did not fall during the first session.
5. We call $X$ the random variable which, for each sample of 100 people who have subscribed to the formula, associates the number of them who did not fall during either the first or the second session. We assimilate the choice of a sample of 100 people to a draw with replacement. We admit that the probability that a person does not fall during either the first or the second session is 0.24.
   1. [a.] Show that the random variable $X$ follows a binomial distribution whose parameters you will specify.
   2. [b.] What is the probability of having, in a sample of 100 people who have subscribed to the formula, at least 20 people who do not fall during either the first or the second session?
   3. [c.] Calculate the expectation $E(X)$ and interpret the result in the context of the exercise.