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 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.

Exercise 2

5 points With a concern for environmental preservation, Mr. Durand decides to go to work each morning using his bicycle or public transport. If he chooses to take public transport one morning, he takes public transport again the next day with a probability equal to 0.8. If he uses his bicycle one morning, he uses his bicycle again the next day with a probability equal to 0.4. For every non-zero natural number $n$, we denote:

• $T _ { n }$ the event ``Mr. Durand uses public transport on the $n$-th day''
• $V _ { n }$ the event ``Mr. Durand uses his bicycle on the $n$-th day''
• We denote $p _ { n }$ the probability of the event $T _ { n }$,

On the first morning, he decides to use public transport. Thus, the probability of the event $T _ { 1 }$ is $p _ { 1 } = 1$.

1. Copy and complete the probability tree below representing the situation for the $2 ^ { \mathrm { nd } }$ and $3 ^ { \mathrm { rd } }$ days.
2. Calculate $p _ { 3 }$
3. On the $3 ^ { \mathrm { rd } }$ day, Mr. Durand uses his bicycle. Calculate the probability that he took public transport the day before.
4. Copy and complete the probability tree below representing the situation for the $n$-th and ( $n + 1$ )-th days.
5. Show that, for every non-zero natural number $n$, $p _ { n + 1 } = 0,2 p _ { n } + 0,6$.
6. Show by induction that, for every non-zero natural number $n$, we have $$p _ { n } = 0,75 + 0,25 \times 0,2 ^ { n - 1 } .$$
7. Determine the limit of the sequence ( $p _ { n }$ ) and interpret the result in the context of the exercise.

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. No justification is required. A wrong answer, a multiple answer or the absence of an answer to a question neither awards nor deducts points.
We consider L a list of numbers consisting of consecutive terms of an arithmetic sequence with first term 7 and common difference 3, the last number in the list is 2023, namely: $$\mathrm{L} = [7, 10, \ldots, 2023].$$
Question 1: The number of terms in this list is:

Answer A	Answer B	Answer C	Answer D
2023	673	672	2016

Question 2: We choose a number at random from this list. The probability of drawing an even number is:

Answer A	Answer B	Answer C	Answer D
$\frac{1}{2}$	$\frac{34}{673}$	$\frac{336}{673}$	$\frac{337}{673}$

We choose a number at random from this list. We are interested in the following events:

• Event $A$: ``obtain a multiple of 4''
• Event $B$: ``obtain a number whose units digit is 6''

We are given $p(A \cap B) = \frac{34}{673}$.
Question 3: The probability of obtaining a multiple of 4 having 6 as the units digit is:

Answer A	Answer B	Answer C	Answer D
$\frac{168}{673} \times \frac{34}{673}$	$\frac{34}{673}$	$\frac{17}{84}$	$\frac{168}{34}$

Question 4: $P_B(A)$ is equal to:

Answer A	Answer B	Answer C	Answer D
$\frac{36}{168}$	$\frac{1}{2}$	$\frac{33}{168}$	$\frac{34}{67}$

Question 5: We choose, at random, successively, 10 elements from this list. An element can be chosen multiple times. The probability that none of these 10 numbers is a multiple of 4 is:

\begin{tabular}{ c } Answer A

$\left(\frac{505}{673}\right)^{10}$

&

Answer B

$1 - \left(\frac{505}{673}\right)^{10}$

&

Answer C

$\left(\frac{168}{673}\right)^{10}$

&

Answer D

$1 - \left(\frac{168}{673}\right)^{10}$

\hline \end{tabular}

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. $p = 1 - ( 1 - x ) ^ { n }$

Alice has two urns A and B each containing four indistinguishable balls. Urn A contains two green balls and two red balls. Urn B contains three green balls and one red ball. Alice randomly chooses an urn and then a ball from that urn. She obtains a green ball. The probability that she chose urn B is:
A. $\frac{3}{8}$
B. $\frac{1}{2}$
C. $\frac{3}{5}$
D. $\frac{5}{8}$

A video game rewards players who have won a challenge with a randomly drawn object. The drawn object can be ``common'' or ``rare''. Two types of objects, common or rare, are available: swords and shields.
The video game designers have planned that:

• the probability of drawing a rare object is $7\%$;
• if a rare object is drawn, the probability that it is a sword is $80\%$;
• if a common object is drawn, the probability that it is a sword is $40\%$.

Part A
A player has just won a challenge and draws an object at random. We denote:

• R the event ``the player draws a rare object'';
• $E$ the event ``the player draws a sword'';
• $\bar{R}$ and $\bar{E}$ the complementary events of events $R$ and $E$.

1. Draw a probability tree modelling the situation, then calculate $P(R \cap E)$.
2. Calculate the probability of drawing a sword.
3. The player has drawn a sword. Determine the probability that it is a rare object. Round the result to the nearest thousandth.

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.

The function $f$ is defined on the interval $[ 0 ; 1 ]$ by: $$f ( x ) = \frac { 0,96 x } { 0,93 x + 0,03 }$$
The fight against doping involves carrying out anti-doping tests which aim to determine whether an athlete has used prohibited substances. During a competition bringing together 1000 athletes, a medical team tests all competitors. We propose to study the reliability of this test.
Let $x$ denote the real number between 0 and 1 which represents the proportion of doped athletes. During the development of this test, it was possible to determine that:

• the probability that an athlete is declared positive given that they are doped is equal to 0.96;
• the probability that an athlete is declared positive given that they are not doped is equal to 0.03.

We denote:

• D the event: ``the athlete is doped''.
• $T$ the event: ``the test is positive''.

1. Copy and complete the probability tree.
2. Determine, as a function of $x$, the probability that an athlete is doped and has a positive test.
3. Prove that the probability of event $T$ is equal to $0,93 x + 0,03$.
4. For this question only, assume that there are 50 doped athletes among the 1000 tested. Prove that the probability that an athlete is doped given that their test is positive is equal to $f ( 0,05 )$. Give an approximate value rounded to the nearest hundredth.
5. The positive predictive value of a test is called the probability that the athlete is truly doped when the test result is positive.
   1. [a.] Determine from which value of $x$ the positive predictive value of the test studied will be greater than or equal to 0.9. Round the result to the nearest hundredth.
   2. [b.] A competition official decides to no longer test all athletes, but to target the most successful athletes who are assumed to be more frequently doped. What is the consequence of this decision on the positive predictive value of the test? Argue using a result from Part A.

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?

Exercise 2

We have two opaque urns $\mathrm { U } _ { 1 }$ and $\mathrm { U } _ { 2 }$. Urn $\mathrm { U } _ { 1 }$ contains 4 black balls and 6 white balls. Urn $\mathrm { U } _ { 2 }$ contains 1 black ball and 3 white balls. Consider the following random experiment: We randomly draw a ball from $U _ { 1 }$ which we place in $U _ { 2 }$, then we randomly draw a ball from $\mathrm { U } _ { 2 }$. We denote:

• $N _ { 1 }$ the event ``Drawing a black ball from urn $\mathrm { U } _ { 1 }$''.
• $N _ { 2 }$ the event ``Drawing a black ball from urn $\mathrm { U } _ { 2 }$''.

For any event $A$, we denote $\bar { A }$ its complementary event.
PART A

1. Consider the probability tree opposite. a. Justify that the probability of drawing a black ball from urn $\mathrm { U } _ { 2 }$ given that a white ball was drawn from urn $\mathrm { U } _ { 1 }$ is 0.2. b. Copy and complete the probability tree opposite, showing on each branch the probabilities of the events concerned, in decimal form.
2. Calculate the probability of drawing a black ball from urn $U _ { 1 }$ and a black ball from urn $\mathrm { U } _ { 2 }$.
3. Justify that the probability of drawing a black ball from urn $\mathrm { U } _ { 2 }$ is equal to 0.28.
4. A black ball was drawn from urn $\mathrm { U } _ { 2 }$. Calculate the probability of having drawn a white ball from urn $\mathrm { U } _ { 1 }$. The result will be given in decimal form rounded to $10 ^ { - 2 }$.

PART B $n$ denotes a non-zero natural number. The previous random experiment is repeated $n$ times in an identical and independent manner, that is, urns $\mathrm { U } _ { 1 }$ and $\mathrm { U } _ { 2 }$ are returned to their initial configuration, with respectively 4 black balls and 6 white balls in urn $U _ { 1 }$ and 1 black ball and 3 white balls in urn $\mathrm { U } _ { 2 }$, between each experiment. We denote $X$ the random variable that counts the number of times a black ball is drawn from urn $\mathrm { U } _ { 2 }$. We recall that the probability of drawing a black ball from urn $\mathrm { U } _ { 2 }$ is equal to 0.28 and that of drawing a white ball from urn $\mathrm { U } _ { 2 }$ is equal to 0.72.

1. Determine the probability distribution followed by $X$. Justify your answer.
2. Determine by calculation the smallest natural number $n$ such that: $$1 - 0{,}72 ^ { n } \geqslant 0{,}9$$
3. Interpret the previous result in the context of the experiment.

PART C In this part urns $\mathrm { U } _ { 1 }$ and $\mathrm { U } _ { 2 }$ are returned to their initial configuration, with respectively 4 black balls and 6 white balls in urn $U _ { 1 }$ and 1 black ball and 3 white balls in urn $\mathrm { U } _ { 2 }$.
Consider the following new random experiment: We simultaneously draw two balls from urn $\mathrm { U } _ { 1 }$ which we place in urn $\mathrm { U } _ { 2 }$, then we randomly draw a ball from urn $\mathrm { U } _ { 2 }$.

1. How many possible draws of two balls simultaneously from urn $\mathrm { U } _ { 1 }$ are there?
2. How many possible draws of two balls simultaneously from urn $\mathrm { U } _ { 1 }$ containing exactly one white ball and one black ball are there?
3. Is the probability of drawing a black ball from urn $\mathrm { U } _ { 2 }$ with this new experiment greater than the probability of drawing a black ball from urn $\mathrm { U } _ { 2 }$ with the experiment in part A? Justify your answer. You may use a weighted tree diagram modeling this experiment.

A marketing agency studied customer satisfaction regarding customer service when purchasing a television. These purchases were made either online, in an appliance store chain, or in a large supermarket. Online purchases represent $60 \%$ of sales, appliance store purchases $30 \%$ of sales, and large supermarket purchases $10 \%$ of sales. A survey shows that the proportion of customers satisfied with customer service is:

• $75 \%$ for online customers;
• $90 \%$ for appliance store customers;
• $80 \%$ for large supermarket customers.

A customer who purchased the television model in question is chosen at random. The following events are defined:

• I: ``the customer made their purchase online'';
• $M$: ``the customer made their purchase in an appliance store'';
• $G$: ``the customer made their purchase in a large supermarket'';
• S: ``the customer is satisfied with customer service''.

If $A$ is any event, we denote by $\bar { A }$ its complementary event and $P ( A )$ its probability.

1. Reproduce and complete the tree diagram opposite.
2. Calculate the probability that the customer made their purchase online and is satisfied with customer service.
3. Prove that $P ( S ) = 0.8$.
4. A customer is satisfied with customer service. What is the probability that they made their purchase online? Give the result rounded to $10 ^ { - 3 }$.
5. To conduct the study, the agency must contact 30 customers each day among the television buyers. We assume that the number of customers is large enough to treat the choice of 30 customers as sampling with replacement. Let $X$ be the random variable that, for each sample of 30 customers, associates the number of customers satisfied with customer service. a. Justify that $X$ follows a binomial distribution and specify its parameters. b. Determine the probability, rounded to $10 ^ { - 3 }$, that at least 25 customers are satisfied in a sample of 30 customers contacted on the same day.
6. By solving an inequality, determine the minimum sample size of customers to contact so that the probability that at least one of them is not satisfied is greater than $0.99$.
7. In the two questions a. and b. that follow, we are only interested in online purchases. When a television order is placed by a customer, the delivery time of the television is modeled by a random variable $T$ equal to the sum of two random variables $T _ { 1 }$ and $T _ { 2 }$.

The random variable $T _ { 1 }$ models the integer number of days for the television to be transported from a storage warehouse to a distribution platform. The random variable $T _ { 2 }$ models the integer number of days for the television to be transported from this platform to the customer's home.
We admit that the random variables $T _ { 1 }$ and $T _ { 2 }$ are independent, and we are given:

• The expectation $E \left( T _ { 1 } \right) = 4$ and the variance $V \left( T _ { 1 } \right) = 2$;
• The expectation $E \left( T _ { 2 } \right) = 3$ and the variance $V \left( T _ { 2 } \right) = 1$. a. Determine the expectation $E ( T )$ and the variance $V ( T )$ of the random variable $T$. b. A customer places a television order online. Justify that the probability that they receive their television between 5 and 9 days after their order is greater than or equal to $\frac { 2 } { 3 }$.

During a training session, a volleyball player practises serving. The probability that he succeeds on the first serve is equal to 0.85.
We further assume that the following two conditions are satisfied:

• if the player succeeds on a serve, then the probability that he succeeds on the next one is equal to 0.6;
• if the player fails a serve, then the probability that he fails the next one is equal to 0.6.

For any non-zero natural number $n$, we denote by $R _ { n }$ the event ``the player succeeds on the $n$-th serve'' and $\overline { R _ { n } }$ the complementary event.
Part A We are interested in the first two serves of the training session.

1. Represent the situation with a probability tree.
2. Prove that the probability of event $R _ { 2 }$ is equal to 0.57.
3. Given that the player succeeded on the second serve, calculate the probability that he failed the first one.
4. Let $Z$ be the random variable equal to the number of successful serves during the first two serves. a. Determine the probability distribution of $Z$ (you may use the probability tree from question 1). b. Calculate the mathematical expectation $\mathrm { E } ( Z )$ of the random variable $Z$.

Interpret this result in the context of the exercise.
Part B We now consider the general case. For any non-zero natural number $n$, we denote by $x _ { n }$ the probability of event $R _ { n }$.

1. a. Give the conditional probabilities $P _ { R _ { n } } \left( R _ { n + 1 } \right)$ and $P _ { \overline { R _ { n } } } \left( \overline { R _ { n + 1 } } \right)$. b. Show that, for any non-zero natural number $n$, we have: $x _ { n + 1 } = 0.2 x _ { n } + 0.4$.
2. Let the sequence $(u _ { n })$ be defined for any non-zero natural number $n$ by: $u _ { n } = x _ { n } - 0.5$. a. Show that the sequence $(u _ { n })$ is a geometric sequence. b. Determine the expression of $x _ { n }$ as a function of $n$. Deduce the limit of the sequence $\left( x _ { n } \right)$. c. Interpret this limit in the context of the exercise.

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.
A test has been implemented: this allows to determine (even long after infection), whether or not a person has already been infected with COVID 19. If the test is positive, this means that the person has already been infected with COVID 19.
The sensitivity of a test is the probability that it is positive given that the person has been infected with the disease. The specificity of a test is the probability that the test is negative given that the person has not been infected with the disease.
The test manufacturer provides the following characteristics:

• Its sensitivity is 0.8.
• Its specificity is 0.99.

An individual is drawn and subjected to the test from the adult French population on May 11, 2020. Let $T$ be the event ``the test performed is positive''.

1. Complete the probability tree with the data from the statement.
2. Show that $p(T) = 0.05503$.
3. What is the probability that an individual has been infected given that their test is positive? Give an approximate value to $10^{-4}$ near of the result.

We consider a group from the population of another country subjected to the same test with sensitivity 0.8 and specificity 0.99.
In this group the proportion of individuals with a positive test is 29.44\%.
An individual is chosen at random from this group; what is the probability that they have been infected?

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?

A student eats every day at the university restaurant. This restaurant offers vegetarian and non-vegetarian dishes.

• When on a given day the student has chosen a vegetarian dish, the probability that he chooses a vegetarian dish the next day is 0.9.
• When on a given day the student has chosen a non-vegetarian dish, the probability that he chooses a vegetarian dish the next day is 0.7.

For any natural number $n$, we denote by $V _ { n }$ the event ``the student chose a vegetarian dish on the $n ^ { \mathrm { th } }$ day'' and $p _ { n }$ the probability of $V _ { n }$. On the first day of the semester, the student chose the vegetarian dish. Thus $p _ { 1 } = 1$.

1. a. Indicate the value of $p _ { 2 }$. b. Show that $p _ { 3 } = 0.88$. You may use a probability tree. c. Given that on the 3rd day the student chose a vegetarian dish, what is the probability that he chose a non-vegetarian dish the previous day? Round the result to $10 ^ { - 2 }$.
2. Copy and complete the probability tree.
3. Justify that, for any natural number $n \geqslant 1 , p _ { n + 1 } = 0.2 p _ { n } + 0.7$.
4. We wish to have the list of the first terms of the sequence $( p _ { n } )$ for $n \geqslant 1$. For this, we use a function called meals programmed in Python language, of which three versions are proposed below.

\begin{verbatim} Program 1 def meals(n): p=1 L= [p] for k in range(1,n): p = 0.2*p+0.7 L. append(p) return(L) \end{verbatim}
\begin{verbatim} Program 2 def meals(n): p=1 L= [p] for k in range(1,n+1): p = 0.2*p+0.7 L. append(p) return(L) \end{verbatim}
\begin{verbatim} Program 3 def meals(n): p=1 L=[p] for k in range(1,n): p = 0.2*p+0.7 L.append(p+1) return(L) \end{verbatim}
a. Which of these programs allows displaying the first $n$ terms of the sequence $\left( p _ { n } \right)$? No justification is required. b. With the program chosen in question a., give the result displayed for $n = 5$.
4. Prove by induction that, for any natural number $n \geqslant 1 , p _ { n } = 0.125 \times 0.2 ^ { n - 1 } + 0.875$.
5. Deduce the limit of the sequence $\left( p _ { n } \right)$.

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.

All probabilities, unless otherwise indicated, will be rounded to $10^{-3}$ in this exercise.
A test was developed for the detection of the chikungunya virus. The laboratory manufacturing this test provides the following characteristics:

• the probability that an individual affected by the virus has a positive test is 0.999;
• the probability that an individual not affected by the virus has a positive test is 0.005.

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

• $M$ the event: ``the chosen individual is affected by chikungunya''.
• $T$ the event: ``the test of the chosen individual is positive''.

The test is considered reliable when the probability that an individual with a positive test is affected by the virus is greater than 0.95.

Part A: Study of an example

1. Give the probabilities $P_{M}(T)$ and $P_{\bar{M}}(T)$. ``In March 2005, the epidemic spread rapidly on the island of Réunion, with a major outbreak between late April and early June followed by persistence of viral transmission during the austral winter. In total, 270,000 people were infected out of a total population of 750,000 individuals''. At the end of 2005, the laboratory conducted a mass screening test of the population of the island of Réunion. In this part, the target population is therefore the population of the island of Réunion.
2. Give the exact value of $P(M)$.
3. Copy and complete the weighted tree.
4. Calculate the probability that an individual is affected by the virus and has a positive test.
5. Calculate the probability that an individual has a positive test.
6. Calculate the probability that an individual with a positive test is affected by the virus.
7. Can we estimate that this test is reliable? Argue.

Part B: Screening on a target population

In this part, we denote by $p$ the proportion of people affected by chikungunya virus in a target population. We seek here to test the reliability of this laboratory's test as a function of $p$.

1. Copy, adapting it, the weighted tree from question A3 taking into account the new data.
2. Express the probability $P(T)$ as a function of $p$.
3. Show that $P_{T}(M) = \frac{999p}{994p + 5}$.
4. For which values of $p$ can we consider that this test is reliable?

Part C: Study on a sample

During the epidemic, we assume that the probability of being affected by chikungunya on the island of Réunion is 0.36. We consider a sample of $n$ individuals chosen at random, assimilating this choice to a random draw with replacement. We denote by $X$ the random variable counting the number of infected individuals in this sample among the $n$ drawn. We assume that $X$ follows a binomial distribution with parameters $n$ and $p = 0.36$. Determine from how many individuals $n$ the probability of the event ``at least one of the $n$ inhabitants of this sample is affected by the virus'' is greater than 0.99. Explain the approach.

We have a bag and two urns A and B.

• The bag contains 4 balls: 1 ball with the letter A and 3 balls with the letter B.
• Urn A contains 5 tickets: 3 tickets of 50 euros and 2 tickets of 10 euros.
• Urn B contains 4 tickets: 1 ticket of 50 euros and 3 tickets of 10 euros.

A player randomly draws a ball from the bag:

• if it is a ball with the letter A, he randomly draws a ticket from urn A.
• if it is a ball with the letter B, he randomly draws a ticket from urn B.

We note the following events:

• $A$: the player obtains a ball with the letter A.
• $C$: the player obtains a 50 euro ticket.

1. Copy and complete the tree opposite representing the situation.
2. What is the probability of the event ``the player obtains a ball with the letter A and a ticket of $50 €$''?
3. Prove that the probability $P(C)$ is equal to 0.3375.
4. The player obtained a 10 euro ticket. Is the statement ``There is more than $80\%$ chance that he previously obtained a ball with the letter B'' true? Justify.
5. We denote $X_1$ the random variable that gives the sum, in euros, obtained by the player. Example: if the player obtains a 50 euro ticket, then $X_1 = 50$. Show that the expectation $E(X_1)$ is equal to 23.50 and that the variance $V(X_1)$ is equal to 357.75.
6. After returning the ball to the bag and the ticket to the urn from which it was taken, the player plays a second game. We denote $X_2$ the random variable that gives the sum obtained by the player in this second game. We denote $Y$ the random variable defined as follows: $Y = X_1 + X_2$. a. Show that $E(Y) = 47$. b. Explain why we have $V(Y) = V(X_1) + V(X_2)$.
7. The player plays likewise a third, fourth, \ldots, hundredth game. We thus define in the same way the random variables $X_3, X_4, \ldots, X_{100}$. We denote $Z$ the random variable defined by $Z = X_1 + X_2 + \ldots + X_{100}$. Prove that the probability that $Z$ belongs to the interval $]1950; 2750[$ is greater than or equal to 0.75.