A factory manufactures spherical balls whose diameter is expressed in millimetres. A ball is said to be out of specification when its diameter is less than 9 mm or greater than 11 mm.
Part A
1. Let $X$ be the random variable that associates to each ball chosen at random from production its diameter expressed in mm.
It is assumed that the random variable $X$ follows the normal distribution with mean 10 and standard deviation 0.4.
Show that an approximate value to 0.0001 of the probability that a ball is out of specification is 0.0124. You may use the table of values given in the appendix.
2. A production control is put in place such that 98\% of out-of-specification balls are rejected and 99\% of correct balls are kept.
A ball is chosen at random from production. Let $N$ denote the event: ``the chosen ball is within specification'', and $A$ the event: ``the chosen ball is accepted after the control''.
a. Construct a weighted tree diagram that incorporates the data from the problem statement.
b. Calculate the probability of event $A$.
c. What is the probability that an accepted ball is out of specification?
Part B
This production control proving too costly for the company, it is abandoned: henceforth, all balls produced are kept, and they are packaged in bags of 100 balls.
It is considered that the probability that a ball is out of specification is 0.0124.
It will be assumed that taking a bag of 100 balls at random is equivalent to performing a sampling with replacement of 100 balls from the set of manufactured balls.
Let $Y$ be the random variable that associates to every bag of 100 balls the number of out-of-specification balls in that bag.
1. What is the distribution followed by the random variable $Y$?
2. What are the mean and standard deviation of the random variable $Y$?
3. What is the probability that a bag of 100 balls contains exactly two out-of-specification balls?
4. What is the probability that a bag of 100 balls contains at most one out-of-specification ball?

In a country, the height in centimetres of women aged 18 to 65 can be modelled by a random variable $X _ { 1 }$ following a normal distribution with mean $\mu _ { 1 } = 165 \mathrm {~cm}$ and standard deviation $\sigma _ { 1 } = 6 \mathrm {~cm}$, and that of men aged 18 to 65 by a random variable $X _ { 2 }$ following a normal distribution with mean $\mu _ { 2 } = 175 \mathrm {~cm}$ and standard deviation $\sigma _ { 2 } = 11 \mathrm {~cm}$. In this exercise all results should be rounded to $10 ^ { - 2 }$.

1. What is the probability that a woman chosen at random in this country measures between 1.53 metres and 1.77 metres?
2. a. Determine the probability that a man chosen at random in this country measures more than 1.70 metres. b. Furthermore, it is known that in this country women represent $52 \%$ of the population of people aged between 18 and 65. A person aged between 18 and 65 is chosen at random. They measure more than $1.70 \mathrm {~m}$. What is the probability that this person is a woman?

A company manufactures spherical wooden balls using two production machines A and B. The company considers that a ball can be sold only when its diameter is between $0.9 \mathrm{~cm}$ and $1.1 \mathrm{~cm}$.
Parts A, B and C are independent.
Part A
A study of the operation of the machines made it possible to establish the following results:

• $96\%$ of daily production is saleable.
• Machine A provides $60\%$ of daily production.
• The proportion of saleable balls among the production of machine A is $98\%$.

A ball is chosen at random from the production of a given day. The following events are defined: $A$: ``the ball was manufactured by machine A''; $B$: ``the ball was manufactured by machine B''; $V$: ``the ball is saleable''.

1. Determine the probability that the chosen ball is saleable and comes from machine A.
2. Justify that $P(B \cap V) = 0.372$ and deduce the probability that the chosen ball is saleable given that it comes from machine B.
3. A technician claims that $70\%$ of non-saleable balls come from machine B. Is he correct?

Part B

1. A statistical study leads to modelling the diameter of a ball randomly selected from the production of machine B by a random variable $X$ which follows a normal distribution with mean $\mu = 1$ and standard deviation $\sigma = 0.055$. Verify that the probability that a ball produced by machine B is saleable is indeed that found in Part A, to the nearest hundredth.
2. In the same way, the diameter of a ball randomly selected from the production of machine A is modelled using a random variable $Y$ which follows a normal distribution with mean $\mu = 1$ and standard deviation $\sigma'$, $\sigma'$ being a strictly positive real number. Given that $P(0.9 \leqslant Y \leqslant 1.1) = 0.98$, determine an approximate value to the nearest thousandth of $\sigma'$.

Part C
The saleable balls then pass through a machine that colours them randomly and with equal probability in white, black, blue, yellow or red. After being mixed, the balls are packaged in bags. The quantity produced is large enough that filling a bag can be treated as successive sampling with replacement of balls from daily production.

1. In this question only, the bags are all composed of 40 balls. a. A bag of balls is chosen at random. Determine the probability that the chosen bag contains exactly 10 black balls. Round the result to $10^{-3}$. b. In a bag of 40 balls, 12 black balls were counted. Does this observation allow us to question the adjustment of the machine that colours the balls?
2. If the company wishes the probability of obtaining at least one black ball in a bag to be greater than or equal to $99\%$, what is the minimum number of balls each bag must contain to achieve this objective?

A market gardener specializes in strawberry production.
Part A: strawberry production
The market gardener produces strawberries in two greenhouses denoted A and B; $55\%$ of strawberry flowers are in greenhouse A, and $45\%$ in greenhouse B. In greenhouse A, the probability that each flower produces fruit is equal to 0.88; in greenhouse B, it is equal to 0.84.
For each of the following propositions, indicate whether it is true or false by justifying the answer. An unjustified answer will not be taken into account.
Proposition 1: The probability that a strawberry flower, chosen at random from this farm, produces fruit is equal to 0.862.
Proposition 2: It is observed that a flower, chosen at random from this farm, produces fruit. The probability that it is located in greenhouse A, rounded to the nearest thousandth, is equal to 0.439.
Part B: strawberry packaging
Strawberries are packaged in trays. The mass (expressed in grams) of a tray can be modeled by a random variable $X$ which follows the normal distribution with mean $\mu = 250$ and standard deviation $\sigma$.

1. We are given $P ( X \leqslant 237 ) = 0.14$. Calculate the probability of the event ``the mass of the tray is between 237 and 263 grams''.
2. Let $Y$ be the random variable defined by: $Y = \frac { X - 250 } { \sigma }$. a. What is the distribution of the random variable $Y$? b. Prove that $P \left( Y \leqslant - \frac { 13 } { \sigma } \right) = 0.14$. c. Deduce the value of $\sigma$ rounded to the nearest integer.
3. In this question, we assume that $\sigma$ equals 12. We denote by $n$ and $m$ two integers. a. A tray is compliant if its mass, expressed in grams, lies in the interval $[ 250 - n ; 250 + n ]$. Determine the smallest value of $n$ for a tray to be compliant with a probability greater than or equal to $95\%$. b. In this question, we consider that a tray is compliant if its mass, expressed in grams, lies in the interval $[230; m ]$. Determine the smallest value of $m$ for a tray to be compliant with a probability greater than or equal to $95\%$.

Exercise 1 Common to all candidates

6 POINTS
The three parts are independent. Probability results should be rounded to $10^{-3}$ near.

Part 1

It is estimated that in 2013 the world population consists of 4.6 billion people aged 20 to 79 years and that $46.1\%$ of people aged 20 to 79 years live in rural areas and $53.9\%$ in urban areas. In 2013, according to the International Diabetes Federation, $9.9\%$ of the world population aged 20 to 79 years living in urban areas suffers from diabetes and $6.4\%$ of the world population aged 20 to 79 years living in rural areas suffers from diabetes.
A person aged 20 to 79 years is randomly selected. We denote: $R$ the event: ``the chosen person lives in a rural area'', $D$ the event: ``the chosen person suffers from diabetes''.

1. Translate this situation using a probability tree.
2. a. Calculate the probability that the interviewed person is diabetic. b. The chosen person is diabetic. What is the probability that they live in a rural area?

Part 2

A person is said to be hypoglycemic if their fasting blood glucose is less than $60 \mathrm{mg}.\mathrm{dL}^{-1}$ and they are hyperglycemic if their fasting blood glucose is greater than $110 \mathrm{mg}.\mathrm{dL}^{-1}$. Fasting blood glucose is considered ``normal'' if it is between $70 \mathrm{mg}.\mathrm{dL}^{-1}$ and $110 \mathrm{mg}.\mathrm{dL}^{-1}$. People with a blood glucose level between 60 and $70 \mathrm{mg}.\mathrm{rdL}^{-1}$ are not subject to special monitoring. An adult is randomly chosen from this population. A study established that the probability that they are hyperglycemic is 0.052 to $10^{-3}$ near. In the following, we will assume that this probability is equal to 0.052. We model the fasting blood glucose, expressed in $\mathrm{mg}.\mathrm{dL}^{-1}$, of an adult from a given population, by a random variable $X$ which follows a normal distribution with mean $\mu$ and standard deviation $\sigma$.

1. What is the probability that the chosen person has ``normal'' fasting blood glucose?
2. Determine the value of $\sigma$ rounded to the nearest tenth.
3. In this question, we take $\sigma = 12$. Calculate the probability that the chosen person is hypoglycemic.

Part 3

In order to estimate the proportion, for the year 2013, of people diagnosed with diabetes in the French population aged 20 to 79 years, 10000 people are randomly interviewed. In the sample studied, 716 people were diagnosed with diabetes.

1. Using a confidence interval at the $95\%$ confidence level, estimate the proportion of people diagnosed with diabetes in the French population aged 20 to 79 years.
2. What should be the minimum number of people to interview if we want to obtain a confidence interval with amplitude less than or equal to 0.01?

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?

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\%?

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 distance from a customer's home to a traditional market follows a normal distribution with mean 1740 m and standard deviation 500 m. Among customers whose distance from home to market is 2000 m or more, 15\% use private vehicles to come to the market, and among customers whose distance is less than 2000 m, 5\% use private vehicles. When one customer who came to the market using a private vehicle is randomly selected, what is the probability that the distance from this customer's home to the market is less than 2000 m? (Here, when $Z$ is a random variable following the standard normal distribution, use $\mathrm { P } ( 0 \leqq Z \leqq 0.52 ) = 0.2$ for calculation.) [3 points]
(1) $\frac { 3 } { 8 }$
(2) $\frac { 7 } { 16 }$
(3) $\frac { 1 } { 2 }$
(4) $\frac { 9 } { 16 }$
(5) $\frac { 5 } { 8 }$

The distance from home to a traditional market for customers using a certain traditional market follows a normal distribution with mean 1740 m and standard deviation 500 m. Among customers whose distance from home to the market is 2000 m or more, 15\% use personal vehicles to come to the market, and among customers whose distance is less than 2000 m, 5\% use personal vehicles. When one customer who came to the market using a personal vehicle is randomly selected, what is the probability that the distance from this customer's home to the market is less than 2000 m? (Note: When $Z$ is a random variable following the standard normal distribution, calculate using $\mathrm { P } ( 0 \leqq Z \leqq 0.52 ) = 0.2$.) [3 points]
(1) $\frac { 3 } { 8 }$
(2) $\frac { 7 } { 16 }$
(3) $\frac { 1 } { 2 }$
(4) $\frac { 9 } { 16 }$
(5) $\frac { 5 } { 8 }$

The commute time of employees at a certain company on a certain day follows a normal distribution with mean 66.4 minutes and standard deviation 15 minutes. Among employees whose commute time is 73 minutes or more, 40\% used the subway, and among employees whose commute time is less than 73 minutes, 20\% used the subway, with the remaining employees using other transportation. What is the probability that a randomly selected employee from those who commuted on that day used the subway? (Here, when $Z$ is a random variable following the standard normal distribution, $\mathrm { P } ( 0 \leq Z \leq 0.44 ) = 0.17$.) [4 points]
(1) 0.306
(2) 0.296
(3) 0.286
(4) 0.276
(5) 0.266