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.

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 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.

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.

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.

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?

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.

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 $Y$.

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.

Let $p \in ]0,1[$. We start from the zero matrix of $\mathcal{M}_n(\mathbb{R})$, denoted $M_0$. For all $k \in \mathbb{N}$, we construct the matrix $M_{k+1}$ from the matrix $M_k$ by scanning through the matrix in one pass and changing each zero coefficient to 1 with probability $p$, independently. For $k \geqslant 1$, the number of modifications made during the $k$-th pass is denoted $N_k$.
Give the distribution of $N_1$, then the conditional distribution of $N_2$ given $(N_1 = i)$ for $i$ in a set to be specified. Are $N_1$ and $N_2$ independent?

Let $p \in ]0,1[$, $q = 1-p$, $m = n^2$. For $k \geqslant 1$, the number of modifications made during the $k$-th pass is denoted $N_k$.
Let $r \geqslant 1$ be an integer and $S_r = N_1 + \cdots + N_r$. What does $S_r$ represent? Give its distribution (you may use the previous question).

Exercise V

A six-sided die is rolled five times. Check TRUE if the proposed random variable follows a binomial distribution and FALSE otherwise. V-A- The random variable corresponding to the number of rolls where an even number appears. V-B- The random variable corresponding to the sum of the results of all rolls.

What is the distribution followed by the random variable $A _ { n }$ representing the number of edges of a graph of $\Omega _ { n }$?