We seek to study the number of students knowing the meaning of the acronym URSSAF. For this, we survey them by proposing a multiple choice questionnaire. Each student must choose from three possible answers, denoted $A$, $B$ and $C$, the correct answer being $A$. We denote by $r$ the probability that a student knows the correct answer. Any student knowing the correct answer responds $A$, otherwise they respond at random (equiprobably).

1. We survey a student at random. We denote: $A$ the event ``the student responds $A$'', $B$ the event ``the student responds $B$'', $C$ the event ``the student responds $C$'', $R$ the event ``the student knows the answer'', $\bar{R}$ the complementary event of $R$. a. Translate this situation using a probability tree. b. Show that the probability of event $A$ is $P(A) = \frac{1}{3}(1 + 2r)$. c. Express as a function of $r$ the probability that a person who chose $A$ knows the correct answer.
2. To estimate $r$, we survey 400 people and denote by $X$ the random variable counting the number of correct answers. We will assume that surveying 400 students at random is equivalent to performing sampling with replacement of 400 students from the set of all students. a. Give the distribution of $X$ and its parameters $n$ and $p$ as a function of $r$. b. In an initial survey, we observe that 240 students respond $A$, out of 400 surveyed. Give a 95\% confidence interval for the estimate of $p$. Deduce a 95\% confidence interval for $r$. c. In what follows, we assume that $r = 0.4$. Given the large number of students, we will consider that $X$ follows a normal distribution. i. Give the parameters of this normal distribution. ii. Give an approximate value of $P(X \leqslant 250)$ to $10^{-2}$ precision.

The company claims that $98 \%$ of its standard-sized footballs are compliant with regulations. A check is then carried out on a sample of 250 standard-sized footballs. It is found that 233 of them are compliant with regulations. Does the result of this check call into question the company's claim? Justify your answer. (You may use the confidence interval)

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

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

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

(Candidates who have not followed the specialisation course)
A computer game of chance is set up as follows:

• If the player wins a game, the probability that he wins the next game is $\frac{1}{4}$;
• If the player loses a game, the probability that he loses the next game is $\frac{1}{2}$;
• The probability of winning the first game is $\frac{1}{4}$.

For every non-zero natural number $n$, we denote by $G_{n}$ the event ``the $n^{\mathrm{th}}$ game is won'' and we denote by $p_{n}$ the probability of this event. We thus have $p_{1} = \frac{1}{4}$.

1. Show that $p_{2} = \frac{7}{16}$.
2. Show that, for every non-zero natural number $n$, $p_{n+1} = -\frac{1}{4} p_{n} + \frac{1}{2}$.
3. We thus obtain the first values of $p_{n}$:
   

   $n$	1	2	3	4	5	6	7
   $p_{n}$	0,25	0,4375	0,3906	0,4023	0,3994	0,4001	0,3999

   
   What conjecture can be made?
4. We define, for every non-zero natural number $n$, the sequence $(u_{n})$ by $u_{n} = p_{n} - \frac{2}{5}$. a. Prove that the sequence $(u_{n})$ is a geometric sequence and specify its common ratio. b. Deduce that, for every non-zero natural number $n$, $p_{n} = \frac{2}{5} - \frac{3}{20}\left(-\frac{1}{4}\right)^{n-1}$. c. Does the sequence $(p_{n})$ converge? Interpret this result.

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

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

We randomly interview a viewer. We denote the events:

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

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

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

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

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

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

Exercise 4 (5 points) — Candidates who have followed the specialization course
In a public garden, an artist must install an aquatic artwork. This artwork will consist of two basins A and B as well as a filtering reserve R. Initially, the two basins each contain 100 liters of water. A system of pipes allows the following water transfers to be carried out, every hour and in this order:

• first, half of basin A empties into reserve R;
• then, three quarters of basin B empty into basin A;
• finally, 200 liters of water are added to basin A and 300 liters of water are added to basin B.

The quantities of water in the two basins A and B are modeled using two sequences $(a_n)$ and $(b_n)$: for any natural number $n$, we denote by $a_n$ and $b_n$ the quantities of water in hundreds of liters that will be respectively contained in basins A and B after $n$ hours. For any natural number $n$, we denote by $U_n$ the column matrix $U_n = \binom{a_n}{b_n}$. Thus $U_0 = \binom{1}{1}$.
1. Justify that, for any natural number $n$, $U_{n+1} = MU_n + C$ where $M = \left(\begin{array}{cc} 0.5 & 0.75 \\ 0 & 0.25 \end{array}\right)$ and $C = \binom{2}{3}$.
2. Consider the matrix $P = \left(\begin{array}{cc} 1 & 3 \\ 0 & -1 \end{array}\right)$.
a. Calculate $P^2$. Deduce that the matrix $P$ is invertible and specify its inverse matrix.
b. Show that $PMP$ is a diagonal matrix $D$ that you will specify.
c. Calculate $PDP$.
d. Prove by induction that, for any natural number $n$, $M^n = PD^nP$.
It is admitted henceforth that for any natural number $n$, $M^n = \left(\begin{array}{cc} 0.5^n & 3 \times 0.5^n - 3 \times 0.25^n \\ 0 & 0.25^n \end{array}\right)$.
3. Show that the matrix $X = \binom{10}{4}$ satisfies $X = MX + C$.
4. For any natural number $n$, we define the matrix $V_n$ by $V_n = U_n - X$.
a. Show that for any natural number $n$, $V_{n+1} = MV_n$.
b. It is admitted that, for any non-zero natural number $n$, $V_n = M^n V_0$. Show that for any non-zero natural number $n$, $$U_n = \binom{-18 \times 0.5^n + 9 \times 0.25^n + 10}{-3 \times 0.25^n + 4}.$$
5. a. Show that the sequence $(b_n)$ is increasing and bounded above. Determine its limit.
b. Determine the limit of the sequence $(a_n)$.
c. It is admitted that the sequence $(a_n)$ is increasing. Deduce the capacity of the two basins A and B that must be planned for the feasibility of the project, that is, to avoid any overflow.

In the Pyrenees National Park, a researcher is working on the decline of a protected species in high-mountain lakes: the ``midwife toad''. Parts I and II can be approached independently.

Part I: Effect of the introduction of a new species

In certain lakes in the Pyrenees, trout have been introduced by humans to enable fishing activities in the mountains. The researcher studied the impact of this introduction on the midwife toad population in a lake. His previous studies lead him to model the evolution of this population as a function of time by the following function $f$: $$f ( t ) = \left( 0.04 t ^ { 2 } - 8 t + 400 \right) \mathrm { e } ^ { \frac { t } { 50 } } + 40 \text { for } t \in [ 0 ; 120 ]$$ The variable $t$ represents the elapsed time, in days, from the introduction at time $t = 0$ of trout into the lake, and $f ( t )$ models the number of toads at time $t$.

1. Determine the number of toads present in the lake when the trout are introduced.
2. We admit that the function $f$ is differentiable on the interval $[0 ; 120]$ and we denote $f ^ { \prime }$ its derivative function. Show, by displaying the calculation steps, that for every real number $t$ belonging to the interval $[0 ; 120]$ we have: $$f ^ { \prime } ( t ) = t ( t - 100 ) \mathrm { e } ^ { \frac { t } { 50 } } \times 8 \times 10 ^ { - 4 }$$
3. Study the variations of the function $f$ on the interval $[0 ; 120]$, then draw up the variation table of $f$ on this interval (approximate values to the nearest hundredth will be given).
4. According to this model: a. Determine the number of days $J$ necessary for the number of toads to reach its minimum. What is this minimum number? b. Justify that, after reaching its minimum, the number of toads will one day exceed 140 individuals. c. Using a calculator, determine the duration in days from which the number of toads will exceed 140 individuals.

Part II: Effect of Chytridiomycosis on a tadpole population

One of the main causes of the decline of this toad species in high mountains is a disease, ``Chytridiomycosis'', caused by a fungus. The researcher considers that:

• Three quarters of the mountain lakes in the Pyrenees are not infected by the fungus, that is, they contain no contaminated tadpoles (toad larvae).
• In the remaining lakes, the probability that a tadpole is contaminated is 0.74.

The researcher randomly chooses a lake in the Pyrenees and takes samples from it. For the rest of the exercise, results will be rounded to the nearest thousandth when necessary. The researcher randomly takes a tadpole from the chosen lake to perform a test before releasing it. We denote $T$ the event ``The tadpole is contaminated by the disease'' and $L$ the event ``The lake is infected by the fungus''. We denote $\bar { L }$ the opposite event of $L$ and $\bar { T }$ the opposite event of $T$.

1. Copy and complete the following probability tree using the data from the problem statement.
2. Show that the probability $P ( T )$ that the sampled tadpole is contaminated is 0.185.
3. The tadpole is not contaminated. What is the probability that the lake is infected?

A pharmaceutical laboratory has just developed a new anti-doping test.

Part A

A study on this new test gives the following results:

• if an athlete is doped, the probability that the test result is positive is 0.98 (test sensitivity);
• if an athlete is not doped, the probability that the test result is negative is 0.995 (test specificity).

The test is administered to an athlete selected at random from among the participants in an athletics competition. We denote by $D$ the event ``the athlete is doped'' and $T$ the event ``the test is positive''. We assume that the probability of event $D$ is equal to 0.08.

1. Represent the situation in the form of a probability tree.
2. Prove that $P ( T ) = 0.083$.
3. a. Given that an athlete presents a positive test, what is the probability that he is doped? b. The laboratory decides to market the test if the probability of the event ``an athlete presenting a positive test is doped'' is greater than or equal to 0.95. Will the test proposed by the laboratory be marketed? Justify.

Part B

In a sporting competition, we assume that the probability that a tested athlete presents a positive test is 0.103.

1. In this question 1., we assume that the organizers decide to test 5 athletes selected at random from among the athletes in this competition. We denote by $X$ the random variable equal to the number of athletes presenting a positive test among the 5 tested athletes. a. Give the distribution followed by the random variable $X$. Specify its parameters. b. Calculate the expectation $E ( X )$ and interpret the result in the context of the exercise. c. What is the probability that at least one of the 5 tested athletes presents a positive test?
2. How many athletes must be tested at minimum so that the probability of the event ``at least one tested athlete presents a positive test'' is greater than or equal to 0.75? Justify.

In this exercise, the results of the probabilities requested will be, if necessary, rounded to the nearest thousandth.
Feline leukaemia is a disease affecting cats; it is caused by a virus. In a large veterinary centre, the proportion of cats carrying the disease is estimated at $40\%$. A screening test for the disease is carried out among the cats present in this veterinary centre. This test has the following characteristics.

• When the cat carries the disease, its test is positive in $90\%$ of cases.
• When the cat does not carry the disease, its test is negative in $85\%$ of cases.

A cat is chosen at random from the veterinary centre and the following events are considered:

• $M$: ``The cat carries the disease'';
• $T$: ``The cat's test is positive'';
• $\bar{M}$ and $\bar{T}$ denote the complementary events of events $M$ and $T$ respectively.

1. a. Represent the situation with a probability tree. b. Calculate the probability that the cat carries the disease and that its test is positive. c. Show that the probability that the cat's test is positive is equal to 0.45. d. A cat is chosen from among those whose test is positive. Calculate the probability that it carries the disease.
2. A sample of 20 cats is chosen at random from the veterinary centre. It is assumed that this choice can be treated as sampling with replacement.

Let $X$ be the random variable giving the number of cats presenting a positive test in the chosen sample. a. Determine, by justifying, the distribution followed by the random variable $X$. b. Calculate the probability that there are exactly 5 cats with a positive test in the sample. c. Calculate the probability that there are at most 8 cats with a positive test in the sample. d. Determine the expected value of the random variable $X$ and interpret the result in the context of the exercise.
3. In this question, a sample of $n$ cats is chosen from the centre, which is again treated as sampling with replacement. Let $p_n$ be the probability that there is at least one cat presenting a positive test in this sample. a. Show that $p_n = 1 - 0{,}55^n$. b. Describe the role of the program below written in Python language, in which the variable $n$ is a natural integer and the variable $P$ is a real number. \begin{verbatim} def seuil() : n = 0 P = 0 while P < 0,99 : n = n + 1 P = 1 - 0,55**n return n \end{verbatim} c. Determine, by specifying the method used, the value returned by this program.

In a large French city, electric scooters are made available to users. A company, responsible for maintaining the scooter fleet, checks their condition every Monday.
It is estimated that:

• when a scooter is in good condition on a Monday, the probability that it is still in good condition the following Monday is 0.9;
• when a scooter is in poor condition on a Monday, the probability that it is in good condition the following Monday is 0.4.

We are interested in the condition of a scooter during the inspection phases. Let $n$ be a natural integer. We denote $B_n$ the event ``the scooter is in good condition $n$ weeks after its commissioning'' and $p_n$ the probability of $B_n$. When commissioned, the scooter is in good condition. We therefore have $p_0 = 1$.

1. Give $p_1$ and show that $p_2 = 0.85$. You may rely on a weighted tree.
2. Copy and complete the weighted tree.
3. Deduce that, for all natural integer $n$, $p_{n+1} = 0.5p_n + 0.4$.
4. a. Prove by induction that for all natural integer $n$, $p_n \geqslant 0.8$. b. Based on this result, what communication can the company consider to highlight the reliability of the fleet?
5. a. Consider the sequence $(u_n)$ defined for all natural integer $n$ by $u_n = p_n - 0.8$. Show that $(u_n)$ is a geometric sequence and give its first term and common ratio. b. Deduce the expression of $u_n$ then of $p_n$ as a function of $n$. c. Deduce the limit of the sequence $(p_n)$.

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.

The director of a school wishes to conduct a study among students who took the final examination to analyze how they think they performed on this exam. For this study, students are asked at the end of the exam to answer individually the question: ``Do you think you passed the exam?''.
Only the answers ``yes'' or ``no'' are possible, and it is observed that $91.7\%$ of the students surveyed answered ``yes''. Following the publication of exam results, it is discovered that:

• $65\%$ of students who failed answered ``no'';
• $98\%$ of students who passed answered ``yes''.

A student who took the exam is randomly selected. We denote by $R$ the event ``the student passed the exam'' and $Q$ the event ``the student answered ``yes'' to the question''. For any event $A$, we denote by $P(A)$ its probability and $\bar{A}$ its complementary event.
Throughout the exercise, probabilities are, if necessary, rounded to $10^{-3}$ near.

1. Specify the values of the probabilities $P(Q)$ and $P_{\bar{R}}(\bar{Q})$.
2. Let $x$ be the probability that the randomly selected student passed the exam. a. Copy and complete the weighted tree below. b. Show that $x = 0.9$.
3. The student selected answered ``yes'' to the question. What is the probability that he passed the exam?
4. The grade obtained by a randomly selected student is an integer between 0 and 20. It is assumed to be modeled by a random variable $N$ that follows the binomial distribution with parameters $(20; 0.615)$.
   The director wishes to award a prize to students with the best results.
   Starting from which grade should she award prizes so that $65\%$ of students are rewarded?
5. Ten students are randomly selected.
   The random variables $N_1, N_2, \ldots, N_{10}$ model the grade out of 20 obtained on the exam by each of them. It is admitted that these variables are independent and follow the same binomial distribution with parameters $(20; 0.615)$. Let $S$ be the variable defined by $S = N_1 + N_2 + \cdots + N_{10}$. Calculate the expectation $E(S)$ and the variance $V(S)$ of the random variable $S$.
6. Consider the random variable $M = \frac{S}{10}$. a. What does this random variable $M$ model in the context of the exercise? b. Justify that $E(M) = 12.3$ and $V(M) = 0.47355$. c. Using the Bienaymé-Chebyshev inequality, justify the statement below. ``The probability that the average grade of ten randomly selected students is strictly between 10.3 and 14.3 is at least $80\%$''.

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.

In France there are two formulas for obtaining a driving license:

• Follow supervised driving training from age 15 for 2 years;
• Follow classical training (without supervised driving) from age 17.

In France currently, among young people who follow driving license training, $16\%$ choose supervised driving training, and among them, $74.7\%$ pass the driving test on their first attempt. Following classical training, the success rate on the first attempt is only $56.8\%$.
A young French person who has already taken the driving test is chosen at random, and we consider the following events $A$ and $R$:

• $A$: ``the young person followed supervised driving training'';
• $R$: ``the young person obtained their license on their first attempt''.

Results should be rounded to $10^{-3}$ if necessary.
Part A

1. Draw a probability tree modeling this situation.
2. a. Prove that $P(R) = 0.59664$.
   In the following, we will keep the value 0.597 rounded to $10^{-3}$. b. Give this result as a percentage and interpret it in the context of the exercise.
3. A young person who obtained their license on their first attempt is chosen. What is the probability that they followed supervised driving training?
4. What should be the proportion of young people following supervised driving training if we wanted the overall success rate (regardless of the training chosen) on the first attempt at the driving test to exceed $70\%$?

Part B
A driving school presents 10 candidates for the driving test for the first time, all of whom have followed supervised driving training. The fact of taking driving tests is modeled by independent random trials.
Let $X$ be the random variable giving the number of these 10 candidates who will obtain their license on their first attempt.

1. Justify that $X$ follows a binomial distribution with parameters $n = 10$ and $p = 0.747$.
2. Calculate $P(X \geqslant 6)$. Interpret this result.
3. Determine $E(X)$ and $V(X)$.
4. There are also 40 candidates who have not followed supervised driving training and who are taking the driving test for the first time. In the same way, let $Y$ be the random variable giving the number of these candidates who will obtain their license on their first attempt. We admit that $Y$ is independent of the variable $X$ and that in fact $E(Y) = 22.53$ and $V(Y) = 9.81$.
   Let $Z$ be the random variable counting the total number of candidates (among the 50) who will obtain their driving license on their first attempt at this driving school. a. Express $Z$ as a function of $X$ and $Y$. Deduce $E(Z)$ and $V(Z)$. b. Using Bienaymé-Chebyshev's inequality, show that the probability that there are fewer than 20 or more than 40 candidates who obtain their license on their first attempt is less than 0.12.

Let $n$ be a non-zero natural integer. In the context of a random experiment, we consider a sequence of events $A _ { n }$ and we denote $p _ { n }$ the probability of the event $A _ { n }$. For parts $\mathbf { A }$ and $\mathbf { B }$ of the exercise, we consider that:

• If the event $A _ { n }$ is realized then the event $A _ { n + 1 }$ is realized with probability 0.3.
• If the event $A _ { n }$ is not realized then the event $A _ { n + 1 }$ is realized with probability 0.7.

We assume that $p _ { 1 } = 1$.

Part A:

1. Copy and complete the probabilities on the branches of the probability tree below.
2. Show that $p _ { 3 } = 0.58$.
3. Calculate the conditional probability $P _ { A _ { 3 } } \left( A _ { 2 } \right)$, round the result to $10 ^ { - 2 }$ near.

Part B:

In this part, we study the sequence $( p _ { n } )$ with $n \geqslant 1$.

1. Copy and complete the probabilities on the branches of the probability tree below.
2. a. Show that, for all non-zero natural integer $n$: $p _ { n + 1 } = - 0.4 p _ { n } + 0.7$.

We consider the sequence $( u _ { n } )$, defined for all non-zero natural integer $n$ by: $u _ { n } = p _ { n } - 0.5$. b. Show that $( u _ { n } )$ is a geometric sequence for which we will specify the common ratio and the first term. c. Deduce the expression of $u _ { n }$, then of $p _ { n }$ as a function of $n$. d. Determine the limit of the sequence $\left( p _ { n } \right)$.

Part C:

Let $x \in ] 0 ; 1 [$, we assume that $P _ { \overline { A _ { n } } } \left( A _ { n + 1 } \right) = P _ { A _ { n } } \left( \overline { A _ { n + 1 } } \right) = x$. We recall that $p _ { 1 } = 1$.

1. Show that for all non-zero natural integer $n$: $p _ { n + 1 } = ( 1 - 2 x ) p _ { n } + x$.
2. Prove by induction on $n$ that, for all non-zero natural integer $n$: $$p _ { n } = \frac { 1 } { 2 } ( 1 - 2 x ) ^ { n - 1 } + \frac { 1 } { 2 }$$
3. Show that the sequence $\left( p _ { n } \right)$ is convergent and give its limit.

A screw factory has two machines, I and II, for the production of a certain type of screw.
In September, machine I produced $\frac{54}{100}$ of the total screws produced by the factory. Of the screws produced by this machine, $\frac{25}{1000}$ were defective. In turn, $\frac{38}{1000}$ of the screws produced in the same month by machine II were defective.
The combined performance of the two machines is classified according to the table, in which $P$ indicates the probability of a randomly chosen screw being defective.
$$\begin{aligned} 0 \leq P < \frac{2}{100} & \quad \text{Excellent} \\ \frac{2}{100} \leq P < \frac{4}{100} & \quad \text{Good} \\ \frac{4}{100} \leq P < \frac{6}{100} & \quad \text{Fair} \\ \frac{6}{100} \leq P < \frac{8}{100} & \quad \text{Poor} \\ \frac{8}{100} \leq P \leq 1 & \quad \text{Very Poor} \end{aligned}$$
The combined performance of these machines in September can be classified as
(A) excellent. (B) good. (C) fair. (D) poor. (E) very poor.

A shooting competition is divided into two stages. Each participating team consists of two members. The specific rules are as follows: In the first stage, one team member shoots 3 times. If all 3 shots miss, the team is eliminated with a score of 0. If at least one shot is made, the team advances to the second stage, where the other team member shoots 3 times, earning 5 points for each made shot and 0 points for each missed shot. The team's final score is the total points from the second stage. A participating team consists of members A and B. Let the probability that A makes each shot be $p$, and the probability that B makes each shot be $q$. Each shot is independent.
(1) If $p = 0.4$ and $q = 0.5$, with A participating in the first stage, find the probability that the team's score is at least 5 points.
(2) Assume $0 < p < q$.
(i) To maximize the probability that the team's score is 15 points, who should participate in the first stage?
(ii) To maximize the expected value of the team's score, who should participate in the first stage?

We fix $p, q \in [0,1]$ and $(X_i)_{1 \leq i \leq n}$ a family of $n$ random variables taking values in $\{0,1\}$ mutually independent Bernoulli random variables with parameter $p$. We set $S_n = \sum_{i=1}^n X_i$. We assume in this question that $p < q$.
a. Justify that $$\mathbb{P}\left(\left|\frac{S_n}{n} - q\right| \leq \left|\frac{S_n}{n} - p\right|\right) = \mathbb{P}\left(S_n \geq \frac{p+q}{2}n\right)$$
b. Let $X$ be a Bernoulli random variable with parameter $p$. For $u > 0$, calculate $\mathbb{E}\left(e^{uX}\right)$.
c. Show that for all $u > 0$, $$\mathbb{P}\left(S_n \geq \frac{p+q}{2}n\right) \leq e^{-n\left(\frac{p+q}{2}u - \ln\left(1-p+pe^u\right)\right)}$$ Hint. One may assume that if $(Z_i)_{1 \leq i \leq n}$ are $n$ mutually independent random variables taking a finite number of values, then $\mathbb{E}\left(\prod_{i=1}^n Z_i\right) = \prod_{i=1}^n \mathbb{E}\left(Z_i\right)$.
d. Show that $\mathbb{P}\left(S_n \geq \frac{p+q}{2}n\right) \leq e^{-n\frac{(p-q)^2}{2}}$.

Prove Theorem 3: We fix $p, q \in [0,1]$. Let $n \geq 1$ be an integer and let $S_n$ be a sum of $n$ random variables taking values in $\{0,1\}$ mutually independent Bernoulli random variables with parameter $p$. Then $$\mathbb{P}\left(\left|\frac{S_n}{n} - q\right| \leq \left|\frac{S_n}{n} - p\right|\right) \leq e^{-n\frac{(p-q)^2}{2}}.$$

We set $L_n = \lfloor n^{1/3} \rfloor$, where $\lfloor t \rfloor$ denotes the integer part of a real number $t$. Let $x, y \in \{0,1\}^n$ such that $x \neq y$. Set for $z \in \mathbb{C}$, $A_{x,y}(z) = \sum_{k=0}^{n-1}(x_k - y_k)z^k$.
a. Justify the existence of $\theta_0 \in \left[-\frac{\pi}{L_n}, \frac{\pi}{L_n}\right]$ such that $\left|A_{x,y}\left(e^{i\theta_0}\right)\right| \geq \frac{1}{n^{L_n - 1}}$.
b. Prove that $$\sum_{j=0}^{n-1} \left|\mathbb{E}\left[O_j(x)\right] - \mathbb{E}\left[O_j(y)\right]\right| \cdot \left|\frac{e^{i\theta_0} - (1-p)}{p}\right|^j \geq \frac{p}{n^{L_n - 1}}$$
c. Justify the existence of an integer $j_n(x,y)$ such that $0 \leq j_n(x,y) \leq n-1$ and $$\left|\mathbb{E}\left[O_{j_n(x,y)}(x)\right] - \mathbb{E}\left[O_{j_n(x,y)}(y)\right]\right| \geq \frac{p}{n^{L_n}} \exp\left(-\frac{1-p}{2p^2} \cdot \frac{\pi^2}{L_n^2} n\right).$$

We fix once and for all an integer $n$ which should be considered as being very large. For each pair $(x,y) \in \{0,1\}^n$ such that $x \neq y$, we fix an integer $j_n(x,y)$ whose existence is proved in question 17c. We say that $x$ is better than $y$ given $E^1, E^2, \ldots, E^T$ if $$\left|\frac{1}{T}\sum_{i=1}^T E^i_{j_n(x,y)} - \mathbb{E}\left[O_{j_n(x,y)}(x)\right]\right| < \left|\frac{1}{T}\sum_{i=1}^T E^i_{j_n(x,y)} - \mathbb{E}\left[O_{j_n(x,y)}(y)\right]\right|$$ We set $R_{n,T}(E^1, E^2, \ldots, E^T) = x$ if for all $y \neq x$, $x$ is better than $y$. If we cannot find such an $x$ we set $R_{n,T}(E^1, E^2, \ldots, E^T) = (0,0,\ldots,0)$.
Prove that if $T_n \geq e^{3\ln(n)n^{1/3}}$ then for all $x \in \{0,1\}^n$ and any sequence $$O^1(x), O^2(x), \ldots, O^{T_n}(x)$$ of $T_n$ random variables taking values in $\{0,1\}^n$ mutually independent with the same distribution as $O(x)$, we have $$\max_{x \in \{0,1\}^n} \mathbb{P}\left(R_{n,T_n}\left(O^1(x), O^2(x), \ldots, O^{T_n}(x)\right) \neq x\right) \leq u_n$$ where $(u_n)_{n \geq 1}$ is a sequence tending to 0 as $n$ tends to infinity.
Hint. One may start by writing, justifying it, that $$\mathbb{P}\left(R_{n,T}\left(O^1(x), O^2(x), \ldots, O^T(x)\right) \neq x\right) \leq \sum_{y \in \{0,1\}^n, y \neq x} \mathbb{P}\left(x \text{ is not better than } y \text{ given } O^1(x), O^2(x), \ldots, O^T(x)\right)$$

Show that if $p _ { n } = o \left( \frac { 1 } { n ^ { 2 } } \right)$ in the neighborhood of $+ \infty$, then $\lim _ { n \rightarrow + \infty } \mathbf { P } \left( A _ { n } > 0 \right) = 0$.

Show that if $\frac { 1 } { n ^ { 2 } } = \mathrm { o} \left( p _ { n } \right)$ in the neighborhood of $+ \infty$, then $\lim _ { n \rightarrow + \infty } \mathbf { P } \left( A _ { n } > 0 \right) = 1$.

Deduce a property $\mathcal { P } _ { n }$ and its associated threshold function.

31. The probabilities that a student passes in Mathematics, Physics and Chemistry are $\mathrm { m } , \mathrm { p }$ and c , respectively. Of these subjects, the student has a $75 \%$ chance of passing in atleast one, a $50 \%$ chance of passing in atleast two, and a $40 \%$ chance of passing in exactly two. Which of the following relations are true?
(A) $\mathrm { p } + \mathrm { m } + \mathrm { c } = 19 / 20$
(B) $p + m + c = 27 / 20$
(C) $\mathrm { pmc } = 1 / 10$
(D) $\mathrm { pmc } = 1 / 4$