An urn contains 10 indistinguishable balls to the touch, of which 7 are blue and the others are green. Three successive draws are made with replacement. The probability of obtaining exactly two green balls is: a. $\left(\frac{7}{10}\right)^2 \times \frac{3}{10}$ b. $\left(\frac{3}{10}\right)^2$ c. $\binom{10}{2}\left(\frac{7}{10}\right)\left(\frac{3}{10}\right)^2$ d. $\binom{3}{2}\left(\frac{7}{10}\right)\left(\frac{3}{10}\right)^2$

We randomly choose, independently, $n$ machines from the company, where $n$ denotes a non-zero natural integer. We assimilate this choice to a sampling with replacement, and we denote by $X$ the random variable that associates to each batch of $n$ machines the number of defective machines in this batch. We admit that $X$ follows the binomial distribution with parameters $n$ and $p = 0.082$.
We consider an integer $n$ for which the probability that all machines in a batch of size $n$ function correctly is greater than 0.4.
The largest possible value for $n$ is equal to: a. 5 b. 6 c. 10 d. 11

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 B
A player wins 30 challenges. We denote $X$ the random variable corresponding to the number of rare objects the player obtains after winning 30 challenges. The successive draws are considered independent.

1. Determine, by justifying, the probability distribution followed by the random variable $X$. Specify its parameters, as well as its expected value.
2. Determine $P(X < 6)$. Round the result to the nearest thousandth.
3. Determine the largest value of $k$ such that $P(X \geqslant k) \geqslant 0.5$. Interpret the result in the context of the exercise.
4. The video game developers want to offer players the option to buy a ``gold ticket'' which allows them to draw $N$ objects. The probability of drawing a rare object remains $7\%$. The developers would like that by buying a gold ticket, the probability that a player obtains at least one rare object in these $N$ draws is greater than or equal to $0.95$. Determine the minimum number of objects to draw to achieve this objective. Care should be taken to detail the approach used.

Data published on March 1st, 2023 by the Ministry of Ecological Transition on the registration of private vehicles in France in 2022 contain the following information:

• $22.86\%$ of vehicles were new vehicles;
• $8.08\%$ of new vehicles were rechargeable hybrids;
• $1.27\%$ of used vehicles (that is, those that are not new) were rechargeable hybrids.

Throughout the exercise, probabilities will be rounded to the ten-thousandth.
Part I
In this part, we consider a private vehicle registered in France in 2022. We denote:

• $N$ the event ``the vehicle is new'';
• $R$ the event ``the vehicle is a rechargeable hybrid'';
• $\bar{N}$ and $\bar{R}$ the complementary events of $N$ and $R$.

1. Represent the situation with a probability tree.
2. Calculate the probability that this vehicle is new and a rechargeable hybrid.
3. Prove that the value rounded to the ten-thousandth of the probability that this vehicle is a rechargeable hybrid is 0.0283.
4. Calculate the probability that this vehicle is new given that it is a rechargeable hybrid.

Part II
In this part, we choose 500 private rechargeable hybrid vehicles registered in France in 2022. In what follows, we will assume that the probability that such a vehicle is new is equal to 0.65. We treat the choice of these 500 vehicles as a random draw with replacement. We call $X$ the random variable representing the number of new vehicles among the 500 vehicles chosen.

1. We assume that the random variable $X$ follows a binomial distribution. Specify the values of its parameters.
2. Determine the probability that exactly 325 of these vehicles are new.
3. Determine the probability $p(X \geq 325)$ then interpret the result in the context of the exercise.

Part III
We now choose $n$ private rechargeable hybrid vehicles registered in France in 2022, where $n$ denotes a strictly positive natural number. We recall that the probability that such a vehicle is new is equal to 0.65. We treat the choice of these $n$ vehicles as a random draw with replacement.

1. Give the expression as a function of $n$ of the probability $p_n$ that all these vehicles are used.
2. We denote $q_n$ the probability that at least one of these vehicles is new. By solving an inequality, determine the smallest value of $n$ such that $q_n \geqslant 0.9999$.

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 $80\%$''.

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.

Probabilities requested will be expressed as irreducible fractions
Part A We flip a fair coin three times in succession. We denote $X$ the random variable that counts the number of times, out of the three flips, where the coin landed on ``Heads''.

1. Specify the nature and parameters of the probability distribution followed by $X$.
2. Copy and complete the following table giving the probability distribution of $X$

$k$	0	1	2	3
$P ( X = k )$

Part B Here are the rules of a game where the goal is to obtain three coins on ``Heads'' in one or two attempts:

• We flip three fair coins:
• If all three coins landed on ``Heads'', the game is won;
• Otherwise, the coins that landed on ``Heads'' are kept and those that landed on ``Tails'' are flipped again.
• The game is won if we obtain three coins on ``Heads'', otherwise it is lost.

We consider the following events:

• G: ``the game is won''. And for every integer $k$ between 0 and 3, the events:
• $A _ { k } :$ ``$k$ coins landed on ``Heads'' on the first flip''.

1. Prove that $P _ { A _ { 1 } } ( G ) = \frac { 1 } { 4 }$.
2. Copy and complete the probability tree below.
3. Prove that the probability $p$ of winning this game is $p = \frac { 27 } { 64 }$
4. The game has been won. What is the probability that exactly one coin landed on ``Heads'' on the first attempt?
5. How many times must we play this game for the probability of winning at least one game to exceed 0.95?

Exercise 4 (4 points)

The two parts are independent.

A laboratory manufactures a medicine packaged in the form of tablets.

Part A

A quality control, concerning the mass of tablets, showed that $2 \%$ of tablets have non-conforming mass. These tablets are packaged in boxes of 100 chosen at random from the production line. We admit that the conformity of a tablet is independent of that of the others.
We denote by $N$ the random variable that associates to each box of 100 tablets the number of non-conforming tablets in this box.

1. Justify that the random variable $N$ follows a binomial distribution whose parameters you will specify.
2. Calculate the expectation of $N$ and give an interpretation in the context of the exercise.
3. Results will be rounded to $10 ^ { - 3 }$ near. a. Calculate the probability that a box contains exactly three non-conforming tablets. b. Calculate the probability that a box contains at least 95 conforming tablets.
4. The laboratory director wants to modify the number of tablets per box to be able to state: ``The probability that a box contains only conforming tablets is greater than 0.5''. What is the maximum number of tablets a box should contain to meet this criterion? Justify.

Part B

We admit that the masses of tablets are independent of one another. We take a sample of 100 tablets and we denote $M _ { i }$, for $i$ natural integer between 1 and 100, the random variable that gives the mass in grams of the $i$-th tablet sampled. We consider the random variable $S$ defined by: $$S = M _ { 1 } + M _ { 2 } + \ldots + M _ { 100 } .$$ We admit that the random variables $M _ { 1 } , M _ { 2 } , \ldots , M _ { 100 }$ follow the same probability distribution with expectation $\mu = 2$ and standard deviation $\sigma$.

1. Determine $E ( S )$ and interpret the result in the context of the exercise.
2. We denote by $s$ the standard deviation of the random variable $S$. Show that: $s = 10 \sigma$.
3. We wish that the total mass, in grams, of the tablets contained in a box be strictly between 199 and 201 with a probability at least equal to 0.9. a. Show that this condition is equivalent to: $$P ( | S - 200 | \geqslant 1 ) \leqslant 0.1 .$$ b. Deduce the maximum value of $\sigma$ which allows, using the Bienaymé--Chebyshev inequality, to ensure this condition.

In basketball, it is possible to score baskets worth one point, two points or three points.
The coach of a basketball team decides to study the success statistics of his players' shots. He observes that during training, when Victor attempts a three-point shot, he succeeds with a probability of 0.32. During a training session, Victor makes a series of 15 three-point shots. We assume that these shots are independent.
Let $N$ be the random variable giving the number of baskets scored. The results of the requested probabilities should be, if necessary, rounded to the nearest thousandth.

1. We admit that the random variable $N$ follows a binomial distribution. Specify its parameters.
2. Calculate the probability that Victor succeeds in exactly 4 baskets during this series.
3. Determine the probability that Victor succeeds in at most 6 baskets during this series.
4. Determine the expected value of the random variable $N$.
5. Let $T$ be the random variable giving the number of points scored after this series of shots. a. Express $T$ as a function of $N$. b. Deduce the expected value of the random variable $T$. Give an interpretation of this value in the context of the exercise. c. Calculate $P ( 12 \leqslant T \leqslant 18 )$.

At the output of a tennis ball manufacturing factory, a ball is judged to be compliant in $85\%$ of cases.

1. We test successively 20 balls. We consider that the number of balls is large enough to assimilate these tests to sampling with replacement. We denote $X$ the random variable that counts the number of compliant balls among the 20 tested. a. What is the distribution followed by $X$ and what are its parameters? Justify. b. Calculate $P(X \leqslant 18)$. c. What is the probability that at least two balls are not compliant among the 20 balls tested? d. Determine the expectation of $X$.
2. We now test $n$ balls successively. We consider the $n$ tests as a sample of $n$ independent random variables $X$ following the Bernoulli distribution with parameter 0.85. We consider the random variable $$M_n = \sum_{i=1}^{n} \frac{X_i}{n} = \frac{X_1}{n} + \frac{X_2}{n} + \frac{X_3}{n} + \ldots + \frac{X_n}{n}$$ a. Determine the expectation and variance of $M_n$. b. After recalling the Bienaymé-Chebyshev inequality, show that, for every natural integer $n$, $P\left(0.75 < M_n < 0.95\right) \geqslant 1 - \frac{12.75}{n}$. c. Deduce an integer $n$ such that the average number of compliant balls for a sample of size $n$ belongs to the interval $]0.75 ; 0.95[$ with a probability greater than 0.9.

On a given day, an automatic checkout triggers 15 checks. The probability that a check reveals an error is $p = 0.165$. The detection of an error during a check is independent of other checks. We denote $X$ the random variable equal to the number of errors detected during the checks on this day.

1. We admit that the random variable $X$ follows a binomial distribution. Specify its parameters.
2. Determine the probability that exactly 5 errors are detected. The answer will be given rounded to the nearest hundredth.
3. Determine the probability that at least one error is detected. The answer will be given rounded to the nearest hundredth.
4. We wish to modify the number of checks triggered by the checkout so that the probability that at least one error is detected each day is greater than $99\%$. Determine the number of checks that the checkout must trigger each day for this constraint to be satisfied.

Part A - First model
Based on a data sample, we consider an initial modelling:

• each year, the probability that the El Niño phenomenon is dominant is equal to 0.4;
• the occurrence of the El Niño phenomenon occurs independently from one year to the next.

We denote by $X$ the random variable which, over a period of 10 years, associates the number of years in which El Niño is dominant.

1. Justify that $X$ follows a binomial distribution and specify the parameters of this distribution.
2. a. Calculate the probability that, over a period of 10 years, the El Niño phenomenon is dominant in exactly 2 years. b. Calculate $P ( X \leqslant 2 )$. What does this result mean in the context of the exercise?
3. Calculate $E ( X )$. Interpret this result.

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.

Dominique answers a multiple choice questionnaire with 10 questions. For each question, 4 answers are proposed, of which only one is correct. Dominique answers randomly to each of the 10 questions by checking, for each question, exactly one box among the 4. For each question, the probability that he answers correctly is therefore $\frac { 1 } { 4 }$. We denote $X$ the random variable that counts the number of correct answers to this questionnaire.

1. Determine the distribution followed by the random variable $X$ and give the parameters of this distribution.
2. What is the probability that Dominique obtains exactly 5 correct answers? Round the result to $10 ^ { - 4 }$ near.
3. Give the expectation of $X$ and interpret this result in the context of the exercise.
4. We suppose in this question that a correct answer gives one point and an incorrect answer loses 0.5 points. The final score can therefore be negative.

We denote $Y$ the random variable that gives the number of points obtained. a. Calculate $P ( Y = 10 )$, give the exact value of the result. b. From how many correct answers is Dominique's final score positive? Justify. c. Calculate $P ( Y \leqslant 0 )$, give an approximate value to the nearest hundredth. d. Show that $Y = 1.5 X - 5$. e. Calculate the expectation of the random variable $Y$.

Exercise 3

The ``base64'' encoding, used in computing, allows messages and other data such as images to be represented and transmitted using 64 characters: the 26 uppercase letters, the 26 lowercase letters, the digits 0 to 9 and two other special characters. Parts A, B and C are independent.

Part A

In this part, we are interested in sequences of 4 characters in base64. For example, ``gP3g'' is such a sequence. In a sequence, order must be taken into account: the sequences ``m5C2'' and ``5C2m'' are not identical.

1. Determine the number of possible sequences.
2. Determine the number of sequences if we require that the 4 characters are pairwise different.
3. a. Determine the number of sequences containing no uppercase letter A. b. Deduce the number of sequences containing at least one uppercase letter A. c. Determine the number of sequences containing exactly one uppercase letter A. d. Determine the number of sequences containing exactly two uppercase letters A.

Part B

We are interested in the transmission of a sequence of 250 characters from one computer to another. We assume that the probability that a character is incorrectly transmitted is equal to 0.01 and that the transmissions of the different characters are independent of each other. We denote by $X$ the random variable equal to the number of characters incorrectly transmitted.

1. We admit that the random variable $X$ follows the binomial distribution. Give its parameters.
2. Determine the probability that all characters are correctly transmitted. The exact expression will be given, then an approximate value to $10^{-3}$ near.
3. What do you think of the following statement: ``The probability that more than 16 characters are incorrectly transmitted is negligible''?

Part C

We are now interested in the transmission of 4 sequences of 250 characters. We denote by $X_1, X_2, X_3$ and $X_4$ the random variables corresponding to the numbers of characters incorrectly transmitted during the transmission of each of the 4 sequences. We admit that the random variables $X_1, X_2, X_3$ and $X_4$ are independent of each other and follow the same distribution as the random variable $X$ defined in part B. We denote by $S = X_1 + X_2 + X_3 + X_4$. Determine, by justifying, the expectation and the variance of the random variable $S$.

On an avenue there are 10 traffic lights. Due to a system failure, the traffic lights were without control for one hour, and fixed their lights only in green or red. The traffic lights operate independently; the probability of showing green is $\frac{2}{3}$ and of showing red is $\frac{1}{3}$. A person walked the entire avenue during the period of the failure, observing the color of the light of each of these traffic lights. What is the probability that this person observed exactly one signal in green?
(A) $\frac{10 \times 2}{3^{10}}$
(B) $\frac{10 \times 2^{9}}{3^{10}}$
(C) $\frac{2^{10}}{3^{100}}$
(D) $\frac{2^{90}}{3^{100}}$
(E) $\frac{2}{3^{10}}$

A fair die is thrown 100 times in succession. Find probabilities of the following events.
(i) 4 is the outcome of one or more of the first three throws.
(ii) Exactly 2 of the last 4 throws give an outcome divisible by 3 (i.e., outcome 3 or 6).

When a random variable $X$ follows a binomial distribution $\mathrm { B } \left( 100 , \frac { 1 } { 5 } \right)$, what is the standard deviation of the random variable $3 X - 4$? [3 points]
(1) 12
(2) 15
(3) 18
(4) 21
(5) 24

There is a television with channels set from 1 to 100. The currently active channel is 50. When one button is randomly pressed six times, either the channel increase button or the channel decrease button, what is the probability that the channel returns to 50? (Note: Each time a button is pressed, the channel changes by 1.) [4 points]
(1) $\frac { 1 } { 4 }$
(2) $\frac { 5 } { 16 }$
(3) $\frac { 3 } { 8 }$
(4) $\frac { 7 } { 16 }$
(5) $\frac { 1 } { 2 }$

When rolling a die 20 times, let $X$ be the random variable representing the number of times the face 1 appears, and when tossing a coin $n$ times, let $Y$ be the random variable representing the number of times heads appears. Find the minimum value of $n$ such that the variance of $Y$ is greater than the variance of $X$. [4 points]

[Probability and Statistics] A certain math class has 10 groups, each consisting of 3 male students and 2 female students. When 2 people are randomly selected from each group, let $X$ be the random variable representing the number of groups in which only male students are selected. What is the expected value $\mathrm { E } ( X )$ of $X$? (Note: No student belongs to more than one group.) [3 points]
(1) 6
(2) 5
(3) 4
(4) 3
(5) 2

When the trial of simultaneously tossing 2 coins is repeated 10 times, let $X$ be the random variable representing the number of times both coins show heads. Find the variance $\mathrm { V } ( 4 X + 1 )$ of the random variable $4 X + 1$. [3 points]

A random variable $X$ follows a binomial distribution $\mathrm { B } ( 200 , p )$ and the mean of $X$ is 40. What is the variance of $X$? [2 points]
(1) 32
(2) 33
(3) 34
(4) 35
(5) 36

A random variable $X$ follows a binomial distribution $\mathrm{B}(n, p)$. If the mean and standard deviation of the random variable $2X - 5$ are 175 and 12, respectively, what is the value of $n$? [3 points]
(1) 130
(2) 135
(3) 140
(4) 145
(5) 150

A random variable $X$ follows a binomial distribution $\mathrm { B } ( 9 , p )$, and $\{ \mathrm { E } ( X ) \} ^ { 2 } = \mathrm { V } ( X )$. What is the value of $p$? (Here, $0 < p < 1$) [3 points]
(1) $\frac { 1 } { 13 }$
(2) $\frac { 1 } { 12 }$
(3) $\frac { 1 } { 11 }$
(4) $\frac { 1 } { 10 }$
(5) $\frac { 1 } { 9 }$