To conduct a survey, an employee interviews people chosen at random in a shopping mall. He wonders whether at least three people will agree to answer.

1. In this question, we assume that the probability that a person chosen at random agrees to answer is 0.1. The employee interviews 50 people independently. We consider the events: $A$: ``at least one person agrees to answer'' $B$: ``fewer than three people agree to answer'' $C$: ``three or more people agree to answer''. Calculate the probabilities of events $A$, $B$ and $C$. Round to the nearest thousandth.
2. Let $n$ be a natural integer greater than or equal to 3. In this question, we assume that the random variable $X$ which, to any group of $n$ people interviewed independently, associates the number of people who agreed to answer, follows the probability distribution defined by: $$\left\{\begin{array}{l}\text{For every integer } k \text{ such that } 0 \leqslant k \leqslant n-1,\; P(X = k) = \frac{\mathrm{e}^{-a} a^k}{k!}\\\text{and } P(X = n) = 1 - \sum_{k=0}^{n-1} P(X=k)\end{array}\right.$$

A factory of frozen desserts has an automated line to fill ice cream cones. Ice cream cones are packaged individually and then packaged in batches of 2000 for wholesale sale. It is considered that the probability that a cone has any defect before its packaging in bulk is equal to 0.003. We denote by $X$ the random variable which, to each batch of 2000 cones randomly selected from production, associates the number of defective cones present in this batch. It is assumed that the production is large enough that the draws can be assumed to be independent of each other.

1. What is the distribution followed by $X$? Justify the answer and specify the parameters of this distribution.
2. If a customer receives a batch containing at least 12 defective cones, the company then proceeds to exchange it. Determine the probability that a batch is not exchanged; the result will be rounded to the nearest thousandth.

Question 2

In this hypermarket, a computer model is on promotion. A statistical study made it possible to establish that, each time a customer is interested in this model, the probability that they buy it is equal to 0.3. We consider a random sample of ten customers who were interested in this model. The probability that exactly three of them bought a computer of this model has a value rounded to the nearest thousandth of: a. 0.900 b. 0.092 c. 0.002 d. 0.267

Part A

A competitor participates in an archery competition on a circular target. With each shot, the probability that he hits the target is equal to 0.8.

1. The competitor shoots four arrows. It is considered that the shots are independent. Determine the probability that he hits the target at least three times.
2. How many arrows should the competitor plan to shoot in order to hit the target an average of twelve times?

On a tennis court, a ball launcher allows a player to train alone. This device sends balls one by one at a regular rate. The player then hits the ball and the next ball arrives. According to the manufacturer's manual, the ball launcher sends the ball randomly to the right or to the left with equal probability.
Throughout the exercise, results will be rounded to $10 ^ { - 3 }$ near.

Part A

The player is about to receive a series of 20 balls.

1. What is the probability that the ball launcher sends 10 balls to the right?
2. What is the probability that the ball launcher sends between 5 and 10 balls to the right?

Part B

The ball launcher is equipped with a reservoir that can hold 100 balls. Over a sequence of 100 launches, 42 balls were launched to the right. The player then doubts the proper functioning of the device. Are his doubts justified?

Part C

To increase the difficulty, the player configures the ball launcher to give spin to the balls launched. They can be either ``topspin'' or ``slice''. The probability that the ball launcher sends a ball to the right is still equal to the probability that the ball launcher sends a ball to the left. The device settings allow us to state that:

• the probability that the ball launcher sends a topspin ball to the right is 0.24;
• the probability that the ball launcher sends a slice ball to the left is 0.235.

If the ball launcher sends a slice ball, what is the probability that it is sent to the right?

The different sweets in the bags are all coated with a layer of edible wax. This process, which deforms some sweets, is carried out by two machines A and B. When produced by machine A, the probability that a randomly selected sweet is deformed is equal to 0.05.
On a random sample of 50 sweets from machine A, what is the probability, rounded to the nearest hundredth, that at least 2 sweets are deformed?
Answer a: 0.72 Answer b: 0.28 Answer c: 0.54 Answer d: We cannot answer because data is missing

In 2016, in France, law enforcement carried out 9.8 million alcohol screening tests with motorists, and $3.1\%$ of these tests were positive. In a given region, on 15 June 2016, a gendarmerie unit conducted screening on 200 motorists. Statement 2: rounding to the nearest hundredth, the probability that, out of the 200 tests, there were strictly more than 5 positive tests, is equal to 0.59. Indicate whether Statement 2 is true or false, justifying your answer.

A general knowledge test consists of a multiple choice questionnaire (MCQ) with twenty questions. For each one, the subject proposes four possible answers, of which only one is correct. For each question, the candidate must necessarily choose a single answer. This person earns one point for each correct answer and loses no points if their answer is wrong.
We consider three candidates:

• Anselme answers completely at random to each of the twenty questions. In other words, for each of the questions, the probability that he answers correctly is equal to $\frac { 1 } { 4 }$;
• Barbara is somewhat better prepared. We consider that for each of the twenty questions, the probability that she answers correctly is $\frac { 1 } { 2 }$;
• Camille does even better: for each of the questions, the probability that she answers correctly is $\frac { 2 } { 3 }$.

1. We denote $X , Y$ and $Z$ the random variables equal to the scores respectively obtained by Anselme, Barbara and Camille. a. What is the probability distribution followed by the random variable $X$? Justify. b. Using a calculator, give the answer rounded to the nearest thousandth of the probability $P ( X \geqslant 10 )$. In the following, we will admit that $P ( Y \geqslant 10 ) \approx 0.588$ and $P ( Z \geqslant 10 ) \approx 0.962$.
2. We randomly choose the copy of one of these three candidates.

We denote $A , B , C$ and $M$ the events:

• $A$: ``the chosen copy is Anselme's'';
• $B$: ``the chosen copy is Barbara's'';
• $C$: ``the chosen copy is Camille's'';
• $M$: ``the chosen copy obtains a score greater than or equal to 10''.

We observe, after correcting it, that the chosen copy obtains a score greater than or equal to 10 out of 20.
What is the probability that it is Barbara's copy? Give the answer rounded to the nearest thousandth of this probability.

A statistical study established that one in four clients practises surfing.
In a cable car accommodating 80 clients of the resort, the probability rounded to the nearest thousandth that there are exactly 20 clients practising surfing is: a. 0.560 b. 0.25 c. 1 d. 0.103

An urn contains 5 red balls and 3 white balls indistinguishable to the touch.
A ball is drawn from the urn and its colour is noted. This experiment is repeated 4 times, independently, by replacing the ball in the urn each time.
The probability, rounded to the nearest hundredth, of obtaining at least 1 white ball is: Answer A: 0.15 Answer B: 0.63 Answer C: 0.5 Answer D: 0.85

EXERCISE A - Natural logarithm function
Part A:
In a country, a disease affects the population with a probability of 0.05. There is a screening test for this disease. We consider a sample of $n$ people ($n \geqslant 20$) taken at random from the population, assimilated to a draw with replacement. The sample is tested using this method: the blood of these $n$ individuals is mixed, the mixture is tested. If the test is positive, an individual analysis of each person is performed. Let $X_n$ be the random variable that gives the number of analyses performed.

1. Show that $X_n$ takes the values 1 and $(n+1)$.
2. Prove that $P(X_n = 1) = 0.95^n$.

Establish the distribution of $X_n$ by copying on the answer sheet and completing the following table:

$x_i$	1	$n+1$
$P(X_n = x_i)$

1. What does the expectation of $X_n$ represent in the context of the experiment?

Show that $E(X_n) = n + 1 - n \times 0.95^n$.
Part B:

1. Consider the function $f$ defined on $[20;+\infty[$ by $f(x) = \ln(x) + x\ln(0.95)$.

Show that $f$ is decreasing on $[20;+\infty[$.

1. We recall that $\lim_{x\rightarrow+\infty} \frac{\ln x}{x} = 0$. Show that $\lim_{x\rightarrow+\infty} f(x) = -\infty$.
2. Show that $f(x) = 0$ has a unique solution $a$ on $[20;+\infty[$. Give an approximation to 0.1 of this solution.
3. Deduce the sign of $f$ on $[20;+\infty[$.

Part C:
We seek to compare two types of screening. The first method is described in Part A, the second, more classical, consists of testing all individuals. The first method makes it possible to reduce the number of analyses as soon as $E(X_n) < n$. Using Part B, show that the first method reduces the number of analyses for samples containing at most 87 people.

Question 3: In an urn there are 6 black balls and 4 red balls. We perform 10 successive random draws with replacement. What is the probability (to $10^{-4}$ near) of obtaining 4 black balls and 6 red balls?

a. 0.1662

b. 0.4

c. 0.1115

d. 0.8886

In this part, we model the situation as follows:

• the condition of a scooter is independent of that of the others;
• the probability that a scooter is in good condition is equal to 0.8.

We denote $X$ the random variable which, to a batch of 15 scooters, associates the number of scooters in good condition. Since the number of scooters in the fleet is very large, the sampling of 15 scooters can be assimilated to a draw with replacement.

1. Justify that $X$ follows a binomial distribution and specify the parameters of this distribution.
2. Calculate the probability that all 15 scooters are in good condition.
3. Calculate the probability that at least 10 scooters are in good condition in a batch of 15.
4. We admit that $E(X) = 12$. Interpret the result.

A production company is considering whether to schedule a television game show. This game brings together four candidates and takes place in two phases:

• The first phase is a qualification phase. This phase depends only on chance. For each candidate, the probability of qualifying is 0.6.
• The second phase is a competition between the qualified candidates. It only takes place if at least two candidates are qualified. Its duration depends on the number of qualified candidates as indicated in the table below (when there is no second phase, its duration is considered to be zero).

\begin{tabular}{ l } Number of candidates qualified

for the second phase

& 0 & 1 & 2 & 3 & 4 \hline

Duration of the second phase in

minutes

& 0 & 0 & 5 & 9 & 11 \hline \end{tabular}
For the company to decide to retain this game, the following two conditions must be verified: Condition no. 1: The second phase must take place in at least 80\% of cases. Condition no. 2: The average duration of the second phase must not exceed 6 minutes. Can the game be retained?

This exercise is a multiple choice questionnaire. For each question, only one of the four proposed answers is correct. No justification is required.
A wrong answer, multiple answers or the absence of an answer to a question neither awards nor deducts points. The five questions are independent.
A production line produces mechanical parts. It is estimated that $4 \%$ of the parts produced by this line are defective. We randomly choose $n$ parts produced by the production line. The number of parts produced is large enough that this choice can be treated as a draw with replacement. We denote $X$ the random variable equal to the number of defective parts drawn. In the following three questions, we take $n = 50$.

1. What is the probability, rounded to the nearest thousandth, of drawing at least one defective part? a. 1 b. 0,870 c. 0,600 d. 0,599
2. The probability $p ( 3 < X \leqslant 7 )$ is equal to : a. $p ( X \leqslant 7 ) - p ( X > 3 )$ b. $p ( X \leqslant 7 ) - p ( X \leqslant 3 )$ c. $p ( X < 7 ) - p ( X > 3 )$ d. $p ( X < 7 ) - p ( X \geqslant 3 )$
3. What is the smallest natural integer $k$ such that the probability of drawing at most $k$ defective parts is greater than or equal to $95 \%$ ? a. 2 b. 3 c. 4 d. 5

In the following questions, $n$ no longer necessarily equals 50.
4. What is the probability of drawing only defective parts? a. $0,04 ^ { n }$ b. $0,96 ^ { n }$ c. $1 - 0,04 ^ { n }$ d. $1 - 0,96 ^ { n }$
5. Consider the Python function below. What does it return?
\begin{verbatim} def seuil (x) : n=1 while 1-0.96**n a. The smallest number $n$ such that the probability of drawing at least one defective part is greater than or equal to x . b. The smallest number $n$ such that the probability of drawing no defective parts is greater than or equal to x. c. The largest number $n$ such that the probability of drawing only defective parts is greater than or equal to x. d. The largest number $n$ such that the probability of drawing no defective parts is greater than or equal to x.

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 $p = 0.082$.
In this question, we take $n = 50$.
The value of the probability $p(X > 2)$, rounded to the nearest thousandth, is: a. 0.136 b. 0.789 c. 0.864 d. 0.924

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

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.

A store is equipped with self-service automatic checkouts where the customer scans their own items. The checkout software regularly triggers verification requests.
The check can be either ``complete'': the store employee then scans all of the customer's items again; or ``partial'': the employee then selects one or more of the customer's items to verify that they have been scanned correctly.
If a check is triggered, it is a complete check one time out of ten. When a complete check is triggered, a customer error is detected in $30\%$ of cases. When a partial check is performed, in $85\%$ of cases, there is no error.
A check is triggered at an automatic checkout. We consider the following events:

• T: ``The check is a complete check'';
• E: ``An error is detected during the check''.

We denote $\bar{T}$ and $\bar{E}$ the complementary events of $T$ and $E$.

1. Construct a probability tree representing the situation and then determine $P(\bar{T} \cap E)$.
2. Calculate the probability that an error is detected during a check.
3. Determine the probability that a complete check was performed, given that an error was detected. The answer will be given rounded to the nearest hundredth.

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.

The store has three identical automatic checkouts which, during a day, each triggered 20 checks. We denote $X_1, X_2$ and $X_3$ the random variables associating to each checkout the number of errors detected during this day. We admit that the random variables $X_1, X_2$ and $X_3$ are independent of each other and each follow a binomial distribution $\mathscr{B}(20; 0.165)$.

1. Determine the exact values of the expectation and variance of the random variable $X_1$.
2. We define the random variable $S$ by $S = X_1 + X_2 + X_3$.
   Justify that $E(S) = 9.9$ and that $V(S) = 8.2665$. For this question, we will use 10 as the value of $E(S)$. Using the Bienaymé-Chebyshev inequality, show that the probability that the total number of errors on the day is strictly between 6 and 14 is greater than 0.48.