The industrialist claims that only $2\%$ of the valves he manufactures are defective. We assume this claim is true, and we denote $F$ the random variable equal to the frequency of defective valves in a random sample of 400 valves taken from total production.

1. Determine the interval $I$ of asymptotic fluctuation at the $95\%$ threshold of the variable $F$.
2. We choose 400 valves at random from production. We treat this choice as a random draw of 400 valves, with replacement, from production. Among these 400 valves, 10 are defective. In light of this result, can we question, at the $95\%$ threshold, the industrialist's claim?

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)

A study conducted in 2000 showed that the percentage of French people regularly consuming ice cream was 84\%. In 2010, out of 900 people surveyed, 795 of them declared consuming ice cream.
Can we affirm, at the 95\% confidence level and based on the study of this sample, that the percentage of French people regularly consuming ice cream remained stable between 2000 and 2010?

The supplier claims that, among the high-end padlocks, there are no more than $3\%$ of defective padlocks in his production. The manager of the hardware store wishes to verify the validity of this claim in his stock; for this purpose, he takes a random sample of 500 high-end padlocks, and finds 19 that are defective.
Does this check call into question the fact that the stock contains no more than $3\%$ of defective padlocks?
For this, you may use an asymptotic fluctuation interval at the $95\%$ threshold.

In July 2014, the health surveillance institute of an island published that $15\%$ of the population is affected by the virus. To verify whether the actual proportion is higher, a sample of 1000 people chosen at random from this island is studied. The population is large enough to consider that such a sample results from draws with replacement.
We denote by $X$ the random variable which, for any sample of 1000 people chosen at random, corresponds to the number of people affected by the virus and by $F$ the random variable giving the associated frequency.

1. a. Under the hypothesis $p = 0.15$, determine the distribution of $X$. b. In a sample of 1000 people chosen at random from the island, 197 people affected by the virus are counted. What conclusion can be drawn from this observation about the figure of $15\%$ published by the health surveillance institute? Justify. (You may use the calculation of a fluctuation interval at the $95\%$ threshold.)
2. We now consider that the value of $p$ is unknown. Using the sample from question 1.b., propose a confidence interval for the value of $p$, at the $95\%$ confidence level.

A fair coin is flipped 100 times in succession. Which of the intervals below is an asymptotic fluctuation interval at the 95\% confidence level for the frequency of appearance of heads on this coin? a. $[ 0.371 ; 0.637 ]$ b. $[ 0.480 ; 0.523 ]$ c. [0.402; 0.598] d. $[ 0.412 ; 0.695 ]$

A company wishes to obtain an estimate of the proportion of people over 60 years old among its customers, at the 95\% confidence level, with an interval amplitude less than 0.05.
What is the minimum number of customers to survey? a. 400 b. 800 c. 1600 d. 3200

Part C - Satisfaction survey
The company announces a satisfaction rate of $85\%$ for its customers who called and reached an operator.
A consumer association wishes to verify this rate and surveys 1303 people. Among these, 1150 say they are satisfied. What do you think of the satisfaction rate announced by the company?

Market gardener C claims that $80\%$ of the melons in his production are compliant (mass between 900 g and 1200 g). The retailer doubts this claim. He observes that out of 400 melons delivered by this market gardener during one week, only 294 are compliant. Is the retailer right to doubt the claim of market gardener C?

The operator claims that the density of firs in this communal forest is 1 fir for every 2 trees. On a plot, 106 firs were counted in a sample of 200 trees. Does this result call into question the operator's claim?

The municipality of a large city has a stock of DVDs. Among the $6\%$ of defective DVDs in the entire stock, $98\%$ are removed. It is also admitted that among the non-defective DVDs, $92\%$ are kept in stock; the others are removed.
One of the city's media libraries wonders whether the number of defective DVDs it possesses is not abnormally high. To do this, it performs tests on a sample of 150 DVDs from its own stock which is large enough for this sample to be treated as successive sampling with replacement. On this sample, 14 defective DVDs are detected.
The asymptotic fluctuation interval at the $95\%$ threshold is given by the formula $$\left[ p - 1{,}96 \frac{\sqrt{p(1-p)}}{\sqrt{n}} ; p + 1{,}96 \frac{\sqrt{p(1+p)}}{\sqrt{n}} \right]$$ where $n$ denotes the sample size and $p$ the proportion of individuals possessing the characteristic studied in this population. The validity conditions are: $n \geqslant 30$, $np \geqslant 5$, $n(1-p) \geqslant 5$.
Can we reject the hypothesis that in this media library, $6\%$ of DVDs are defective?

In a supermarket, a department manager wishes to develop the supply of organic products. To justify his approach, he claims to his supervisor that $75\%$ of customers buy organic products at least once a month.
The supervisor wishes to verify his claims. To do this, he organizes a survey at the store exit. Of 2000 people interviewed, 1421 respond that they consume organic products at least once a month.
At the $95\%$ confidence level, what can we think of the department manager's claim?

Part C

Louise's company states on its website that $35\%$ of its employees practice carpooling. A survey conducted within the company shows that out of 254 employees randomly selected, 82 practice carpooling. Does this survey call into question the information published by the company on its website?

A customer orders a batch of 400 flutes of $12.5 \mathrm { cL }$ and finds that 13 of them do not conform to the characteristics announced by the manufacturer. The sales manager had nevertheless assured him that $98 \%$ of the flutes sold by the company were compliant. Does the customer's batch allow, at a risk of $5 \%$, to question the sales manager's claim?

Since the reform and opening up, people's payment methods have undergone tremendous changes. In recent years, mobile payment has become one of the main payment methods. To understand the usage of two mobile payment methods, $A$ and $B$, among students at a certain school last month, 100 students were randomly selected from the entire school. It was found that 5 people in the sample used neither method. The distribution of payment amounts for students in the sample who used only method $A$ and only method $B$ is as follows:

Payment Amount (yuan)	$( 0,1000 ]$	$( 1000,2000 ]$	Greater than 2000
Payment Method
Using only $A$	18 people	9 people	3 people

(I) Randomly select 1 student from the entire school. Estimate the probability that this student used both payment methods $A$ and $B$ last month; (II) Randomly select 1 student each from the sample students who used only $A$ and only $B$. Let $X$ denote the number of people among these 2 people whose payment amount last month exceeded 1000 yuan. Find the probability distribution and mathematical expectation of $X$; (III) It is known that the payment methods of sample students did not change this month. Now, 3 students are randomly selected from the sample students who used only method $A$, and it is found that their payment amounts this month all exceeded 2000 yuan. Based on the sampling results, can we conclude that the number of students using only method $A$ in the sample whose payment amount this month exceeded 2000 yuan has changed? Explain the reasoning.

Using the result of Q40, deduce a probabilistic interpretation of the stability condition studied in Part III (i.e., the condition on $r = \frac{\tau}{\delta^2}$ found in Q34).

A message is transmitted over a channel where each bit is inverted independently with probability $1-p \in ]0,1[$. Blocks of $r$ bits are transmitted. $X$ denotes the number of inversions during the transmission of a block of $r$ bits, and $X$ follows the distribution determined in Q16. We consider $\alpha \in ]0,1[$ and the condition $$\mathbb{P}(X \geqslant 2) \leqslant 1 - \alpha. \tag{II.2}$$ Using Markov's inequality, prove that if $r \leqslant 2 \frac{1-\alpha}{1-p}$, then condition (II.2) is satisfied.

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

Let $S$ be the sample space of all $3 \times 3$ matrices with entries from the set $\{ 0,1 \}$. Let the events $E _ { 1 }$ and $E _ { 2 }$ be given by $$\begin{aligned} & E _ { 1 } = \{ A \in S : \operatorname { det } A = 0 \} \text { and } \\ & E _ { 2 } = \{ A \in S : \text { sum of entries of } A \text { is } 7 \} . \end{aligned}$$ If a matrix is chosen at random from $S$, then the conditional probability $P \left( E _ { 1 } \mid E _ { 2 } \right)$ equals $\_\_\_\_$

The probability that a missile hits a target successfully is 0.75. In order to destroy the target completely, at least three successful hits are required. Then the minimum number of missiles that have to be fired so that the probability of completely destroying the target is NOT less than 0.95, is $\_\_\_\_$

Two fair dice, each with faces numbered $1, 2, 3, 4, 5$ and $6$, are rolled together and the sum of the numbers on the faces is observed. This process is repeated till the sum is either a prime number or a perfect square. Suppose the sum turns out to be a perfect square before it turns out to be a prime number. If $p$ is the probability that this perfect square is an odd number, then the value of $14p$ is $\_\_\_\_$

Consider the system of linear equations in two variables $\left\{\begin{array}{c} ax + 6y = 6 \\ x + by = 1 \end{array}\right.$, where the coefficients $a, b$ are determined by rolling a fair die and flipping a fair coin respectively. Let $a$ be the number of points shown on the die; if the coin shows heads, $b = 1$; if the coin shows tails, $b = 2$. Select the correct options.
(1) The probability of rolling $a = b$ is $\frac{1}{3}$
(2) The probability that the system has no solution is $\frac{1}{12}$
(3) The probability that the system has a unique solution is $\frac{5}{6}$
(4) The probability that the coin shows tails and the system has a solution is $\frac{1}{2}$
(5) Given that the coin shows tails and the system has a solution, the probability that $x$ is positive is $\frac{2}{5}$