Exercise 5 — Candidates who have not followed the specialization course In this exercise, all requested probabilities will be rounded to $10 ^ { - 4 }$. We study a model of automobile air conditioner composed of a mechanical module and an electronic module. If a module fails, it is replaced.
Part A: Study of mechanical module failures An automobile maintenance company has found, through a statistical study, that the operating time (in months) of the mechanical module can be modeled by a random variable $D$ that follows a normal distribution with mean $\mu = 50$ and standard deviation $\sigma$:

1. Determine the rounding to $10 ^ { - 4 }$ of $\sigma$ knowing that the statistical service indicates that $P ( D \geqslant 48 ) = 0,7977$.

For the rest of this exercise, we will take $\sigma = 2,4$.

1. Determine the probability that the operating time of the mechanical module is between 45 and 52 months.
2. Determine the probability that the mechanical module of an air conditioner that has been operating for 48 months will continue to function for at least 6 more months.

Part B: Study of electronic module failures On the same air conditioner model, the automobile maintenance company has found that the operating time (in months) of the electronic module can be modeled by a random variable $T$ that follows an exponential distribution with parameter $\lambda$.

1. Determine the exact value of $\lambda$, knowing that the statistical service indicates that $P ( 0 \leqslant T \leqslant 24 ) = 0,03$.

For the rest of this exercise, we will take $\boldsymbol { \lambda } = 0,00127$.

1. Determine the probability that the operating time of the electronic module is between 24 and 48 months.
2. a. Prove that, for all positive real numbers $t$ and $h$, we have: $P _ { T \geqslant t } ( T \geqslant t + h ) = P ( T \geqslant h )$, that is, the random variable $T$ is memoryless. b. The electronic module of the air conditioner has been operating for 36 months. Determine the probability that it will continue to function for the next 12 months.

Part C: Mechanical and electronic failures We admit that the events ( $D \geqslant 48$ ) and ( $T \geqslant 48$ ) are independent. Determine the probability that the air conditioner does not fail before 48 months.
Part D: Special case of a company garage
A garage of the company has studied the maintenance records of 300 air conditioners over 4 years old. It finds that 246 of them have their mechanical module in working order for 4 years. Should this report call into question the result given by the company's statistical service, namely that $P ( D \geqslant 48 ) = 0,7977$? Justify the answer.

Part A
As part of its activity, a company regularly receives quotation requests. The amounts of these quotations are calculated by its secretariat. A statistical study over the past year leads to modelling the amount of quotations by a random variable $X$ which follows the normal distribution with mean $\mu = 2900$ euros and standard deviation $\sigma = 1250$ euros.

1. If a quotation request received by the company is chosen at random, what is the probability that the quotation amount exceeds 4000 euros?
2. In order to improve the profitability of its activity, the entrepreneur decides not to follow up on $10\%$ of requests. He discards those with the lowest quotation amounts. What must be the minimum amount of a requested quotation for it to be taken into account? Give this amount to the nearest euro.

We study the production of a factory that manufactures sweets, packaged in bags. A bag is chosen at random from daily production. The mass of this bag, expressed in grams, is modelled by a random variable $X$ which follows a normal distribution with mean $\mu = 175$. Furthermore, statistical observation has shown that $2\%$ of bags have a mass less than or equal to 170 g, which is expressed in the model considered by: $P ( X \leqslant 170 ) = 0.02$.
What is the probability, rounded to the nearest hundredth, of the event ``the mass of the bag is between 170 and 180 grams''?
Answer a: 0.04 Answer b: 0.96 Answer c: 0.98 Answer d: We cannot answer because data is missing.

In an automobile factory, certain metal parts are covered with a thin layer of nickel that protects them against corrosion and wear. The process used is electroplating with nickel.
It is assumed that the random variable $X$, which associates to each treated part the thickness of nickel deposited, follows a normal distribution with mean $\mu _ { 1 } = 25$ micrometers ( $\mu \mathrm { m }$ ) and standard deviation $\sigma _ { 1 }$.
A part is compliant if the thickness of nickel deposited is between $22.8 \mu \mathrm {~m}$ and $27.2 \mu \mathrm {~m}$.
The probability density function of $X$ is represented below. It was determined that $P ( X > 27.2 ) = 0.023$.

1. a. Determine the probability that a part is compliant. b. Justify that 1.1 is an approximate value of $\sigma _ { 1 }$ to within $10 ^ { - 1 }$. c. Given that a part is compliant, calculate the probability that the thickness of nickel deposited on it is less than $24 \mu \mathrm {~m}$. Round to $10 ^ { - 3 }$.
2. A team of engineers proposes another nickel plating process, obtained by chemical reaction without any current source. The team claims that this new process theoretically allows obtaining $98 \%$ of compliant parts. The random variable $Y$ which, for each part treated with this new process, associates the thickness of nickel deposited follows a normal distribution with mean $\mu _ { 2 } = 25 \mu \mathrm {~m}$ and standard deviation $\sigma _ { 2 }$. a. Assuming the above claim, compare $\sigma _ { 1 }$ and $\sigma _ { 2 }$. b. A quality control evaluates the new process; it reveals that out of 500 parts tested, 15 are not compliant. At the $95 \%$ confidence level, can we reject the team's claim?

Exercise 3 (3 points)

Part A:

A health control agency is interested in the number of bacteria of a certain type contained in fresh cream. It performs analyses on 10000 samples of 1 ml of fresh cream from the entire French production. The results are given in the table below:

\begin{tabular}{ l } Number of bacteria

(in thousands)

& $[100;120[$ & $[120;130[$ & $[130;140[$ & $[140;150[$ & $[150;160[$ & $[160;180[$ \hline Number of samples & 1597 & 1284 & 2255 & 1808 & 1345 & 1711 \hline \end{tabular}
Using a calculator, give an estimate of the mean and standard deviation of the number of bacteria per sample.

Part B:

The agency then decides to model the number of bacteria studied (in thousands per ml) present in fresh cream by a random variable $X$ following the normal distribution with parameters $\mu = 140$ and $\sigma = 19$.

1. a. Is this choice of modelling relevant? Argue. b. We denote $p = P(X \geqslant 160)$. Determine the value of $p$ rounded to $10^{-3}$.
   
2. During the inspection of a dairy, the health control agency analyzes a sample of 50 samples of 1 ml of fresh cream from the production of this dairy; 13 samples contain more than 160 thousand bacteria. a. The agency declares that there is an anomaly in the production and that it can affirm it with a probability of 0.05 of being wrong. Justify its declaration. b. Could it have affirmed it with a probability of 0.01 of being wrong?

Exercise 3

All requested results will be rounded to the nearest thousandth.

1. A study conducted on a population of men aged 35 to 40 years showed that the total cholesterol level in the blood, expressed in grams per liter, can be modeled by a random variable $T$ that follows a normal distribution with mean $\mu = 1.84$ and standard deviation $\sigma = 0.4$. a. Determine according to this model the probability that a subject randomly selected from this population has a cholesterol level between $1.04\mathrm{~g/L}$ and $2.64\mathrm{~g/L}$. b. Determine according to this model the probability that a subject randomly selected from this population has a cholesterol level greater than $1.2\mathrm{~g/L}$.
   
2. In order to test the effectiveness of a cholesterol-lowering drug, patients needing treatment agreed to participate in a clinical trial organized by a laboratory. In this trial, $60\%$ of patients took the drug for one month, the others taking a placebo (neutral tablet). We study the decrease in cholesterol level after the experiment.
   A decrease in this level is observed in $80\%$ of patients who took the drug. No decrease is observed in $90\%$ of people who took the placebo. A patient who participated in the experiment is randomly selected and we denote:
   • $M$ the event ``the patient took the drug'';
   • $B$ the event ``the patient's cholesterol level decreased''.
   a. Translate the data from the statement using a probability tree. b. Calculate the probability of event $B$. c. Calculate the probability that a patient took the drug given that their cholesterol level decreased.
   
3. The laboratory that produces this drug announces that $30\%$ of patients who use it experience side effects. To test this hypothesis, a cardiologist randomly selects 100 patients treated with this drug. a. Determine the asymptotic confidence interval at the $95\%$ threshold for the proportion of patients undergoing this treatment and experiencing side effects. b. The study conducted on 100 patients counted 37 people experiencing side effects. What can we conclude? c. To estimate the proportion of users of this drug experiencing side effects, an independent organization conducts a study based on a confidence interval at the $95\%$ confidence level. This study results in an observed frequency of $37\%$ of patients experiencing side effects, and a confidence interval that does not contain the frequency $30\%$. What is the minimum sample size for this study?

The mass in grams of melons from market gardener B is modelled by a random variable $M_\mathrm{B}$ that follows a normal distribution with mean 1050 and unknown standard deviation $\sigma$. Melons are described as ``compliant'' if their mass is between 900 g and 1200 g. The retailer observes that $85\%$ of melons supplied by market gardener B are compliant. Determine the standard deviation $\sigma$ of the random variable $M_\mathrm{B}$. Give the value rounded to the nearest integer.

Exercise 1 (6 points)

We study certain characteristics of a supermarket in a small town.

Part A - Preliminary Demonstration

Let $X$ be a random variable that follows the exponential distribution with parameter 0.2. Recall that the expectation of the random variable $X$, denoted $E(X)$, is equal to: $$\lim_{x \rightarrow +\infty} \int_{0}^{x} 0.2t\, \mathrm{e}^{-0.2t} \mathrm{~d}t$$ The purpose of this part is to demonstrate that $E(X) = 5$.

1. Let $g$ be the function defined on the interval $[0; +\infty[$ by $g(t) = 0.2t\,\mathrm{e}^{-0.2t}$.
   We define the function $G$ on the interval $[0; +\infty[$ by $G(t) = (-t-5)\mathrm{e}^{-0.2t}$. Verify that $G$ is a primitive of $g$ on the interval $[0; +\infty[$.
2. Deduce that the exact value of $E(X)$ is 5.
   Hint: you may use, without proving it, the following result: $$\lim_{x \rightarrow +\infty} x\,\mathrm{e}^{-0.2x} = 0$$

Part B - Study of the duration of a customer's presence in the supermarket

A study commissioned by the supermarket manager makes it possible to model the duration, expressed in minutes, spent in the supermarket by a randomly chosen customer using a random variable $T$. This variable $T$ follows a normal distribution with expectation 40 minutes and standard deviation a positive real number denoted $\sigma$. Thanks to this study, it is estimated that $P(T < 10) = 0.067$.

1. Determine an approximate value of the real number $\sigma$ to the nearest second.
2. In this question, we take $\sigma = 20$ minutes. What is then the proportion of customers who spend more than one hour in the supermarket?

Part C - Waiting time for payment

This supermarket gives customers the choice to use automatic payment terminals alone or to go through a checkout managed by an operator.

1. The waiting time at an automatic terminal, expressed in minutes, is modeled by a random variable that follows the exponential distribution with parameter $0.2\,\mathrm{min}^{-1}$. a. Give the average waiting time for a customer at an automatic payment terminal. b. Calculate the probability, rounded to $10^{-3}$, that the waiting time for a customer at an automatic payment terminal is greater than 10 minutes.
2. The study commissioned by the manager leads to the following modeling:
   • among customers who chose to use an automatic terminal, 86\% wait less than 10 minutes;
   • among customers using a checkout, 63\% wait less than 10 minutes.
   We randomly choose a customer from the store and define the following events: $B$: ``the customer pays at an automatic terminal''; $\bar{B}$: ``the customer pays at a checkout with an operator''; $S$: ``the customer's waiting time during payment is less than 10 minutes''.
   A waiting time greater than ten minutes at a checkout with an operator or at an automatic terminal creates a negative perception of the store in the customer. The manager wants more than 75\% of customers to wait less than 10 minutes. What is the minimum proportion of customers who must choose an automatic payment terminal for this objective to be achieved?

Part D - Gift vouchers

During payment, scratch cards, winning or losing, are distributed to customers. The number of cards distributed depends on the amount of purchases. Each customer receives one scratch card per 10~\euro{} of purchases. For example, if the purchase amount is 58.64~\euro{}, then the customer receives 5 cards; if the amount is 124.31~\euro{}, the customer receives 12 cards. Winning cards represent 0.5\% of the entire stock of cards. Furthermore, this stock is large enough to treat the distribution of a card as a draw with replacement.

1. A customer makes purchases for an amount of 158.02~\euro{}.
   What is the probability, rounded to $10^{-2}$, that they obtain at least one winning card?
2. From what purchase amount, rounded to 10~\euro{}, is the probability of obtaining at least one winning card greater than 50\%?

The number of trees per hectare in this forest can be modelled by a random variable $X$ following a normal distribution with mean $\mu = 4000$ and standard deviation $\sigma = 300$.

1. Determine the probability that there are between 3400 and 4600 trees on a given hectare of this forest. The result will be given rounded to $10^{-3}$.
2. Calculate the probability that there are more than 4500 trees on a given hectare of this forest. The result will be given rounded to $10^{-3}$.

Part of the city's DVD stock consists of animated films intended for young audiences. An animated film is chosen at random and we denote by $X$ the random variable that gives the duration, in minutes, of this film. $X$ follows a normal distribution with mean $\mu = 80$ min and standard deviation $\sigma$. Furthermore, it is estimated that $P(X \geqslant 92) = 0{,}10$.

1. Determine the real number $\sigma$ and give an approximate value to 0.01.
2. A child watches an animated film whose duration he does not know. Knowing that he has already watched one hour and thirty minutes, what is the probability that the film ends within the next five minutes?

In parts A and B of this exercise, we consider a disease; every individual has an equal probability of 0.15 of being affected by this disease.

Part A

This part is a multiple choice questionnaire (M.C.Q.). For each question, only one of the four answers is correct. A correct answer earns one point, an incorrect answer or no answer earns or deducts no points.
A screening test for this disease has been developed. If the individual is sick, in 94\% of cases the test is positive. For an individual chosen at random from this population, the probability that the test is positive is 0.158.

1. An individual chosen at random from the population is tested: the test is positive. A value rounded to the nearest hundredth of the probability that the person is sick is equal to : A: 0.94 B: 1 C: 0.89
   D : we cannot know
2. A random sample is taken from the population, and the test is administered to individuals in this sample. We want the probability that at least one individual tests positive to be greater than or equal to 0.99. The minimum sample size must be equal to : A: 26 people B: 27 people C: 3 people D: 7 people
3. A vaccine to fight this disease has been developed. It is manufactured by a company in the form of a dose injectable by syringe. The volume $V$ (expressed in millilitres) of a dose follows a normal distribution with mean $\mu = 2$ and standard deviation $\sigma$. The probability that the volume of a dose, expressed in millilitres, is between 1.99 and 2.01 millilitres is equal to 0.997. The value of $\sigma$ must satisfy : A: $\sigma = 0.02$
   B : $\sigma < 0.003$ C: $\sigma > 0.003$
   D : $\sigma = 0.003$

Part B

1. A box of a certain medicine can cure a sick person.

The duration of effectiveness (expressed in months) of this medicine is modelled as follows:

• during the first 12 months after manufacture, it is certain to remain effective;
• beyond that, its remaining duration of effectiveness follows an exponential distribution with parameter $\lambda$.

The probability that one of the boxes taken at random from a stock has a total duration of effectiveness greater than 18 months is equal to 0.887. What is the average value of the total duration of effectiveness of this medicine?
2. A city of 100,000 inhabitants wants to build up a stock of these boxes in order to treat sick people. What must be the minimum size of this stock so that the probability that it is sufficient to treat all sick people in this city is greater than 95\%?

A company packages white sugar from two farms $U$ and $V$ in 1 kg packets of different qualities. Extra fine sugar is packaged separately in packets bearing the label ``extra fine''. Throughout the exercise, results should be rounded, if necessary, to the nearest thousandth.
To calibrate the sugar according to the size of its crystals, it is passed through a series of three sieves. Sugar crystals with a size less than $0.2\,\mathrm{mm}$ are found in the sealed bottom container at the end of calibration. They will be packaged in packets bearing the label ``extra fine sugar''.

1. A sugar crystal is randomly selected from farm U. The size of this crystal, expressed in millimeters, is modeled by the random variable $X_{\mathrm{U}}$ which follows the normal distribution with mean $\mu_{\mathrm{U}} = 0.58\,\mathrm{mm}$ and standard deviation $\sigma_{\mathrm{U}} = 0.21\,\mathrm{mm}$. a. Calculate the probabilities of the following events: $X_{\mathrm{U}} < 0.2$ and $0.5 \leqslant X_{\mathrm{U}} < 0.8$. b. 1800 grams of sugar from farm $U$ is passed through the series of sieves. Deduce from the previous question an estimate of the mass of sugar recovered in the sealed bottom container and an estimate of the mass of sugar recovered in sieve 2.
2. A sugar crystal is randomly selected from farm V. The size of this crystal, expressed in millimeters, is modeled by the random variable $X_{\mathrm{V}}$ which follows the normal distribution with mean $\mu_{\mathrm{V}} = 0.65\,\mathrm{mm}$ and standard deviation $\sigma_{\mathrm{V}}$ to be determined. During the calibration of a large quantity of sugar crystals from farm V, it is observed that $40\%$ of these crystals end up in sieve 2. What is the value of the standard deviation $\sigma_{\mathrm{V}}$ of the random variable $X_{\mathrm{V}}$?

A factory manufactures tubes.
Part A
Questions 1. and 2. are independent. We are interested in two types of tubes, called type 1 tubes and type 2 tubes.

1. A type 1 tube is accepted at inspection if its thickness is between 1.35 millimetres and 1.65 millimetres. a. Let $X$ denote the random variable that, for each type 1 tube randomly selected from the day's production, gives its thickness expressed in millimetres. We assume that the random variable $X$ follows a normal distribution with mean 1.5 and standard deviation 0.07.
   A type 1 tube is randomly selected from the day's production. Calculate the probability that the tube is accepted at inspection. b. The company wishes to improve the quality of type 1 tube production. To do this, the settings of the machines producing these tubes are modified. Let $X _ { 1 }$ denote the random variable that, for each type 1 tube selected from the production of the modified machine, gives its thickness. We assume that the random variable $X _ { 1 }$ follows a normal distribution with mean 1.5 and standard deviation $\sigma _ { 1 }$.
   A type 1 tube is randomly selected from the production of the modified machine. Determine an approximate value to $10 ^ { - 3 }$ of $\sigma _ { 1 }$ so that the probability that this tube is accepted at inspection equals 0.98. (You may use the random variable $Z$ defined by $Z = \frac { X _ { 1 } - 1.5 } { \sigma _ { 1 } }$ which follows the standard normal distribution.)
   
2. A machine produces type 2 tubes. A type 2 tube is said to be ``compliant for length'' when its length, in millimetres, belongs to the interval [298; 302]. The specifications establish that, in the production of type 2 tubes, a proportion of $2 \%$ of tubes that are not ``compliant for length'' is acceptable.
   It is desired to decide whether the production machine should be serviced. To do this, a random sample of 250 tubes is taken from the production of type 2 tubes, in which 10 tubes are found to be not ``compliant for length''. a. Give an asymptotic confidence interval at $95 \%$ for the frequency of tubes not ``compliant for length'' in a sample of 250 tubes. b. Should the machine be serviced? Justify your answer.

Part B
Adjustment errors in the production line can affect the thickness or length of type 2 tubes.
A study conducted on the production revealed that:

• $96 \%$ of type 2 tubes have compliant thickness;
• among type 2 tubes that have compliant thickness, $95 \%$ have compliant length;
• $3.6 \%$ of type 2 tubes have non-compliant thickness and compliant length.

A type 2 tube is randomly selected from the production and we consider the events: --- $E$: ``the thickness of the tube is compliant''; --- $L$: ``the length of the tube is compliant''. We model the random experiment with a probability tree.

1. Copy and complete this tree entirely.
2. Show that the probability of event $L$ equals 0.948.

The standard inner diameter of a bearing on a roller wheel is 8 mm. Let $X$ denote the random variable giving in mm the diameter of a bearing and we assume that $X$ follows a normal distribution with mean 8 and standard deviation 0.1.
A bearing is said to be compliant if its diameter is between $7.8 \mathrm{~mm}$ and $8.2 \mathrm{~mm}$.

1. Calculate the probability that a bearing is compliant.
2. Supplier $B$ sells its bearings in batches of 16 and claims that only $5\%$ of its bearings are non-compliant. The club president, who bought 30 batches from him, finds that 38 bearings are non-compliant. Does this check call into question supplier B's claim? An asymptotic fluctuation interval at the $95\%$ threshold may be used.
3. The bearing manufacturer of this supplier decides to improve the production of its bearings. The adjustment of the machine that manufactures them is modified so that $96\%$ of the bearings are compliant. We assume that after adjustment the random variable $X$ follows a normal distribution with mean 8 and standard deviation $\sigma$. a. What is the distribution followed by $\frac{X - 8}{\sigma}$? b. Determine $\sigma$ so that the manufactured bearing is compliant with a probability equal to 0.96.

A company specializes in the sale of tiles.
Parts A, B and C are independent.

Part A

We assume in this part that the company sells batches of tiles containing $25\%$ of tiles with pattern and $75\%$ of white tiles. During a quality control, it is observed that:

• $2.25\%$ of the tiles are cracked;
• $6\%$ of the tiles with pattern are cracked.

A tile is randomly selected. We denote by $M$ the event ``the tile has a pattern'' and $F$ the event ``the tile is cracked''.

1. Translate the situation using a probability tree.
2. We know that the selected tile is cracked. Prove that the probability that it is a tile with pattern is $\frac{2}{3}$.
3. Calculate $P_{\bar{M}}(F)$, the probability of $F$ given $\bar{M}$.

Part B

We model the thickness in millimeters of a randomly selected tile by a random variable $X$ that follows a normal distribution with mean $\mu = 11$ and standard deviation $\sigma$.
A tile is marketable if its thickness measures between $10.1\text{ mm}$ and $11.9\text{ mm}$. We know that $99\%$ of the tiles are marketable.

1. Prove that $P(X < 10.1) = 0.005$.
2. We introduce the random variable $Z$ such that $$Z = \frac{X - 11}{\sigma}.$$ a. Give the distribution followed by the random variable $Z$. b. Prove that $P\left(Z \leqslant -\frac{0.9}{\sigma}\right) = 0.005$. c. Deduce the value of $\sigma$ rounded to the nearest hundredth.

Part C

We consider the function $f$ defined on $[0; 2\pi]$ by $$f(x) = -1.5\cos(x) + 1.5$$ We admit that the function $f$ is continuous on $[0; 2\pi]$. We denote by $\mathscr{C}_1$ the representative curve of the function $f$ in an orthonormal coordinate system.

1. Prove that the function $f$ is positive on $[0; 2\pi]$.
2. In the figure above, the curve drawn in dashes, denoted $\mathscr{C}_2$, is the curve symmetric to $\mathscr{C}_1$ with respect to the $x$-axis. The shape of a tile is that of the region bounded by the curves $\mathscr{C}_1$ and $\mathscr{C}_2$. We denote by $\mathscr{A}$ its area, expressed in square units. Calculate $\mathscr{A}$.

In this exercise and unless otherwise stated, results should be rounded to $10^{-3}$.
A factory manufactures tubes.

Part A

Questions 1. and 2. are independent. We are interested in two types of tubes, called type 1 tubes and type 2 tubes.

1. A type 1 tube is accepted at inspection if its thickness is between 1.35 millimetres and 1.65 millimetres. a. Let $X$ denote the random variable which, for each type 1 tube randomly selected from the day's production, gives its thickness expressed in millimetres. We assume that the random variable $X$ follows a normal distribution with mean 1.5 and standard deviation 0.07.
   A type 1 tube is randomly selected from the day's production. Calculate the probability that the tube is accepted at inspection. b. The company wishes to improve the quality of type 1 tube production. To do this, the settings of the machines producing these tubes are modified. Let $X_1$ denote the random variable which, for each type 1 tube selected from the production of the modified machine, gives its thickness. We assume that the random variable $X_1$ follows a normal distribution with mean 1.5 and standard deviation $\sigma_1$.
   A type 1 tube is randomly selected from the production of the modified machine. Determine an approximate value to $10^{-3}$ of $\sigma_1$ so that the probability that this tube is accepted at inspection equals 0.98. (You may use the random variable $Z$ defined by $Z = \frac{X_1 - 1.5}{\sigma_1}$ which follows the standard normal distribution.)
   
2. A machine produces type 2 tubes. A type 2 tube is said to be ``compliant for length'' when its length, in millimetres, belongs to the interval [298; 302]. The specifications establish that, in the production of type 2 tubes, a proportion of $2\%$ of tubes that are not ``compliant for length'' is acceptable.
   It is desired to decide whether the production machine should be serviced. To do this, a random sample of 250 tubes is taken from the production of type 2 tubes, in which 10 tubes are found to be not ``compliant for length''. a. Give an asymptotic confidence interval at $95\%$ for the frequency of tubes not ``compliant for length'' in a sample of 250 tubes. b. Should the machine be serviced? Justify your answer.

Part B

Adjustment errors in the production line can affect the thickness or length of type 2 tubes.
A study conducted on the production revealed that: --- $96\%$ of type 2 tubes have compliant thickness; --- among type 2 tubes that have compliant thickness, $95\%$ have compliant length; --- $3.6\%$ of type 2 tubes have non-compliant thickness and compliant length.
A type 2 tube is randomly selected from the production and we consider the events: --- $E$: ``the tube's thickness is compliant''; --- $L$: ``the tube's length is compliant''.
We model the random experiment with a probability tree.

1. Copy and complete this tree entirely.
2. Show that the probability of event $L$ equals 0.948.

During a professional examination, each candidate must present a file of type A or a file of type B; $60\%$ of candidates present a file of type A, the others presenting a file of type B. The jury assigns to each file a mark between 0 and 20. A candidate passes if the mark assigned to their file is greater than or equal to 10. A file is chosen at random. It is admitted that the mark assigned to a file of type A can be modeled by a random variable $X$ following the normal distribution with mean 11.3 and standard deviation 3, and the mark assigned to a file of type B by a random variable $Y$ following the normal distribution with mean 12.4 and standard deviation 4.7. We may denote $A$ the event: ``the file is a file of type A'', $B$ the event: ``the file is a file of type B'', and $R$ the event: ``the file is that of a candidate who passed the examination''. Probabilities will be rounded to the nearest hundredth.

1. The chosen file is of type A. What is the probability that this file is that of a candidate who passed the examination? It is admitted that the probability that the chosen file, given that it is of type B, is that of a candidate who passed is equal to 0.70.
2. Show that the probability, rounded to the nearest hundredth, that the chosen file is that of a candidate who passed the examination is equal to 0.68.
3. The jury examines 500 files chosen randomly from files of type B. Among these files, 368 are those of candidates who passed the examination.
   A jury member claims that this sample is not representative. He justifies his claim by explaining that in this sample, the proportion of candidates who passed is too large. What argument can be put forward to confirm or contest his claims?
4. The jury awards a ``jury prize'' to files that obtained a mark greater than or equal to $N$, where $N$ is an integer. The probability that a file chosen at random obtains the ``jury prize'' is between 0.10 and 0.15. Determine the integer $N$.

The maximum thickness of an avalanche, expressed in centimetres, can be modelled by a random variable $X$ which follows a normal distribution with mean $\mu = 150 \mathrm{~cm}$ and unknown standard deviation. We know that $P ( X \geqslant 200 ) = 0.025$. What is the probability $P ( X \geqslant 100 )$ ? a. We cannot b. 0.025 c. 0.95 d. 0.975 answer because there are missing elements in the problem statement.

Part B

The travel time for Louise, in minutes, between her home and work, can be modeled by a random variable $X$ that follows a normal distribution with mean 28 and standard deviation 5.

1. Calculate $P ( X \leqslant 25 )$.
2. Calculate the probability that the travel time is between 18 and 38 minutes.
3. Determine the travel duration $d$, rounded to the nearest minute, such that $P ( X \geqslant d ) = 0.1$.
4. Louise has now found a faster route. From now on, the travel time, in minutes, can be modeled by a random variable $Y$ that follows a normal distribution with mean 26 and standard deviation $\sigma$. We know that $P ( Y \geqslant 30 ) = 0.1$. Determine $\sigma$ rounded to the nearest hundredth.

The red-billed tropicbird is a bird of intertropical regions.
1. When the red-billed tropicbird lives in a polluted environment, its lifespan, in years, is modelled by a random variable $X$ following a normal distribution with unknown mean $\mu$ and standard deviation $\sigma = 0.95$.
a. Consider the random variable $Y$ defined by $Y = \frac { X - \mu } { 0.95 }$.
Give without justification the distribution followed by the variable $Y$.
b. It is known that $P ( X \geqslant 4 ) = 0.146$.
Prove that the value of $\mu$ rounded to the nearest integer is 3.
2. When the red-billed tropicbird lives in a healthy environment, its lifespan, in years, is modelled by a random variable $Z$.
The curves of the density functions associated with the distributions of $X$ and $Z$ are represented in the APPENDIX to be returned with the answer sheet.
a. Which is the curve of the density function associated with $X$? Justify.
b. On the APPENDIX to be returned with the answer sheet, shade the region of the plane corresponding to $P ( Z \geqslant 4 )$.
It will be admitted henceforth that $P ( Z \geqslant 4 ) = 0.677$.
3. A statistical study of a given region established that $30\%$ of red-billed tropicbirds live in a polluted environment; the others live in a healthy environment.
A red-billed tropicbird living in the given region is chosen at random.
Consider the following events:

• $S$ : ``the red-billed tropicbird chosen lives in a healthy environment'';
• $V$ : ``the red-billed tropicbird chosen has a lifespan of at least 4 years''.

a. Complete the weighted tree illustrating the situation on the APPENDIX to be returned with the answer sheet.
b. Determine $P ( V )$. Round the result to the nearest thousandth.
c. Given that the red-billed tropicbird has a lifespan of at least 4 years, what is the probability that it lives in a healthy environment? Round the result to the nearest thousandth.

A machine manufactures balls intended for a game of chance. The mass in grams of each of these balls can be modeled by a random variable $M$ following a normal distribution with mean 52 and standard deviation $\sigma$. Balls whose mass is between 51 and 53 grams are said to be compliant.

1. With the initial settings of the machine we have $\sigma = 0.437$. Under these conditions, calculate the probability that a ball manufactured by this machine is compliant. An approximate value to $10 ^ { - 1 }$ near the result will be given.
2. It is considered that the machine is correctly adjusted if at least $99 \%$ of the balls it manufactures are compliant. Determine an approximate value of the largest value of $\sigma$ that allows us to affirm that the machine is correctly adjusted.

A music club is preparing for its regular concert this year. Based on past experience, the attendance rate among invited guests is 0.5. When 100 people are randomly selected from the invited guests,

$z$	$\mathrm { P } ( 0 \leqq Z \leqq z )$
1.0	0.3413
1.2	0.3849
1.4	0.4192
1.6	0.4452

what is the probability that the attendance rate is at least 0.43 and at most 0.56, using the standard normal distribution table on the right? [4 points]
(1) 0.8041
(2) 0.7698
(3) 0.7605
(4) 0.7262
(5) 0.6826

Suppose the weight of products produced at a certain factory follows a normal distribution $\mathrm { N } \left( 11,2 ^ { 2 } \right)$. Two people, $A$ and $B$, each independently extracted a sample of size 4. Using the standard normal distribution table on the right, what is the probability that the sample means extracted by both $A$ and $B$ are between 10 and 14 inclusive? [3 points]

$z$	$\mathrm { P } ( 0 \leqq Z \leqq z )$
1	0.3413
2	0.4772
3	0.4987

(1) 0.8123
(2) 0.7056
(3) 0.6587
(4) 0.5228
(5) 0.2944

The weight of products manufactured at a certain factory follows a normal distribution $\mathrm { N } \left( 11,2 ^ { 2 } \right)$. Two people A and B each independently randomly extracted a sample of size 4. Using the standard normal distribution table on the right, what is the probability that the sample means of both A and B are between 10 and 14 inclusive? [3 points]

$z$	$\mathrm { P } ( 0 \leqq Z \leqq z )$
1	0.3413
2	0.4772
3	0.4987

(1) 0.8123
(2) 0.7056
(3) 0.6587
(4) 0.5228
(5) 0.2944

At a certain car wash, the time required to wash one car follows a normal distribution with mean 30 minutes and standard deviation 2 minutes. When washing one car at this car wash, what is the probability that the washing time is 33 minutes or more, using the standard normal distribution table below? [3 points]

$z$	$\mathrm { P } ( 0 \leqq Z \leqq z )$
0.5	0.1915
1.0	0.3413
1.5	0.4332
2.0	0.4772

(1) 0.0228
(2) 0.0668
(3) 0.1587
(4) 0.2708
(5) 0.3085