Question 1

In a hypermarket, $75\%$ of customers are women. One woman in five buys an item from the DIY section, whereas seven men in ten do so.
A person, chosen at random, has made a purchase from the DIY section. The probability that this person is a woman has a value rounded to the nearest thousandth of: a. 0.750 b. 0.150 c. 0.462 d. 0.700

Parts A and B can be treated independently.

Part A

A pharmaceutical laboratory offers screening tests for various diseases. Its communications department highlights the following characteristics:

• the probability that a sick person tests positive is 0.99;
• the probability that a healthy person tests positive is 0.001.

1. For a disease that has just appeared, the laboratory develops a new test. A statistical study makes it possible to estimate that the percentage of sick people among the population of a metropolis is equal to $0.1 \%$. A person is chosen at random from this population and undergoes the test. We denote by $M$ the event ``the chosen person is sick'' and $T$ the event ``the test is positive''. a. Translate the statement in the form of a weighted tree. b. Prove that the probability $p ( T )$ of event $T$ is equal to $$1.989 \times 10 ^ { - 3 } .$$ c. Is the following statement true or false? Justify your answer. Statement: ``If the test is positive, there is less than one chance in two that the person is sick''.
2. The laboratory decides to market a test as soon as the probability that a person who tests positive is sick is greater than or equal to 0.95. We denote by $x$ the proportion of people affected by a certain disease in the population. From what value of $x$ does the laboratory market the corresponding test?

Part B

The laboratory's production line manufactures, in very large quantities, tablets of a medicine.

1. A tablet is compliant if its mass is between 890 and 920 mg. We assume that the mass in milligrams of a tablet taken at random from production can be modeled by a random variable $X$ that follows the normal distribution $\mathscr { N } \left( \mu , \sigma ^ { 2 } \right)$, with mean $\mu = 900$ and standard deviation $\sigma = 7$. a. Calculate the probability that a tablet drawn at random is compliant. Round to $10 ^ { - 2 }$. b. Determine the positive integer $h$ such that $P ( 900 - h \leqslant X \leqslant 900 + h ) \approx 0.99$ to within $10 ^ { - 3 }$.
2. The production line has been adjusted to obtain at least $97 \%$ compliant tablets. To evaluate the effectiveness of the adjustments, a check is performed by taking a sample of 1000 tablets from production. The size of the production is assumed to be large enough that this sample can be treated as 1000 successive draws with replacement. The check made it possible to count 53 non-compliant tablets in the sample taken. Does this check call into question the adjustments made by the laboratory? An asymptotic fluctuation interval at the $95 \%$ threshold can be used.

A pharmaceutical laboratory has just developed a new anti-doping test.

Part A

A study on this new test gives the following results:

• if an athlete is doped, the probability that the test result is positive is 0.98 (test sensitivity);
• if an athlete is not doped, the probability that the test result is negative is 0.995 (test specificity).

The test is administered to an athlete selected at random from among the participants in an athletics competition. We denote by $D$ the event ``the athlete is doped'' and $T$ the event ``the test is positive''. We assume that the probability of event $D$ is equal to 0.08.

1. Represent the situation in the form of a probability tree.
2. Prove that $P ( T ) = 0.083$.
3. a. Given that an athlete presents a positive test, what is the probability that he is doped? b. The laboratory decides to market the test if the probability of the event ``an athlete presenting a positive test is doped'' is greater than or equal to 0.95. Will the test proposed by the laboratory be marketed? Justify.

Part B

In a sporting competition, we assume that the probability that a tested athlete presents a positive test is 0.103.

1. In this question 1., we assume that the organizers decide to test 5 athletes selected at random from among the athletes in this competition. We denote by $X$ the random variable equal to the number of athletes presenting a positive test among the 5 tested athletes. a. Give the distribution followed by the random variable $X$. Specify its parameters. b. Calculate the expectation $E ( X )$ and interpret the result in the context of the exercise. c. What is the probability that at least one of the 5 tested athletes presents a positive test?
2. How many athletes must be tested at minimum so that the probability of the event ``at least one tested athlete presents a positive test'' is greater than or equal to 0.75? Justify.

A manufacturing line produces mechanical parts. It is estimated that $5 \%$ of the parts produced by this line are defective.
An engineer has developed a test to apply to the parts. This test has two possible results: ``positive'' or ``negative''. This test is applied to a part chosen at random from the production of the line. We denote $p ( E )$ the probability of an event $E$. We consider the following events:

• $D$: ``the part is defective'';
• T: ``the part shows a positive test'';
• $\bar { D }$ and $\bar { T }$ denote respectively the complementary events of $D$ and $T$.

Given the characteristics of the test, we know that:

• The probability that a part shows a positive test given that it is defective is equal to 0.98;
• the probability that a part shows a negative test given that it is not defective is equal to 0.97.

PART I

1. Represent the situation using a probability tree.
2. a. Determine the probability that a part chosen at random from the production line is defective and shows a positive test. b. Prove that: $p ( T ) = 0.0775$.
3. The positive predictive value of the test is called the probability that a part is defective given that the test is positive. A test is considered effective if it has a positive predictive value greater than 0.95.

Calculate the positive predictive value of this test and specify whether it is effective.

PART II

A sample of 20 parts is chosen from the production line, treating this choice as a draw with replacement. Let $X$ be the random variable that gives the number of defective parts in this sample. Recall that: $p ( D ) = 0.05$.

1. Justify that $X$ follows a binomial distribution and determine the parameters of this distribution.
2. Calculate the probability that this sample contains at least one defective part.

Give a result rounded to the nearest hundredth.
3. Calculate the expected value of the random variable $X$ and interpret the result obtained.

In this exercise, the results of the probabilities requested will be, if necessary, rounded to the nearest thousandth.
Feline leukaemia is a disease affecting cats; it is caused by a virus. In a large veterinary centre, the proportion of cats carrying the disease is estimated at $40\%$. A screening test for the disease is carried out among the cats present in this veterinary centre. This test has the following characteristics.

• When the cat carries the disease, its test is positive in $90\%$ of cases.
• When the cat does not carry the disease, its test is negative in $85\%$ of cases.

A cat is chosen at random from the veterinary centre and the following events are considered:

• $M$: ``The cat carries the disease'';
• $T$: ``The cat's test is positive'';
• $\bar{M}$ and $\bar{T}$ denote the complementary events of events $M$ and $T$ respectively.

1. a. Represent the situation with a probability tree. b. Calculate the probability that the cat carries the disease and that its test is positive. c. Show that the probability that the cat's test is positive is equal to 0.45. d. A cat is chosen from among those whose test is positive. Calculate the probability that it carries the disease.
2. A sample of 20 cats is chosen at random from the veterinary centre. It is assumed that this choice can be treated as sampling with replacement.

Let $X$ be the random variable giving the number of cats presenting a positive test in the chosen sample. a. Determine, by justifying, the distribution followed by the random variable $X$. b. Calculate the probability that there are exactly 5 cats with a positive test in the sample. c. Calculate the probability that there are at most 8 cats with a positive test in the sample. d. Determine the expected value of the random variable $X$ and interpret the result in the context of the exercise.
3. In this question, a sample of $n$ cats is chosen from the centre, which is again treated as sampling with replacement. Let $p_n$ be the probability that there is at least one cat presenting a positive test in this sample. a. Show that $p_n = 1 - 0{,}55^n$. b. Describe the role of the program below written in Python language, in which the variable $n$ is a natural integer and the variable $P$ is a real number. \begin{verbatim} def seuil() : n = 0 P = 0 while P < 0,99 : n = n + 1 P = 1 - 0,55**n return n \end{verbatim} c. Determine, by specifying the method used, the value returned by this program.

A company receives numerous emails daily. Among these emails, $8\%$ are ``spam'', that is, emails with advertising or malicious intent, which it is desirable not to open. An email received by the company is chosen at random. The properties of the email software used in the company allow us to state that:

• The probability that the chosen email is classified as ``undesirable'' given that it is spam is equal to 0.9.
• The probability that the chosen email is classified as ``undesirable'' given that it is not spam is equal to 0.01.

We denote:

• S the event ``the chosen email is spam'';
• I the event ``the chosen email is classified as undesirable by the email software''.
• $\bar{S}$ and $\bar{I}$ the complementary events of $S$ and $I$ respectively.

1. Model the situation studied using a probability tree, on which the probabilities associated with each branch should appear.
2. a. Prove that the probability that the chosen email is a spam message and is classified as undesirable is equal to 0.072. b. Calculate the probability that the chosen message is classified as undesirable. c. The chosen message is classified as undesirable. What is the probability that it is actually a spam message? Give the answer rounded to the nearest hundredth.
3. A random sample of 50 emails is chosen from those received by the company. It is assumed that this choice amounts to a random draw with replacement of 50 emails from the set of all emails received by the company. Let $Z$ be the random variable counting the number of spam emails among the 50 chosen. a. What is the probability distribution followed by the random variable $Z$, and what are its parameters? b. What is the probability that, among the 50 chosen emails, at least two are spam? Give the answer rounded to the nearest hundredth.

A test is developed to detect a disease in a country. According to the health authorities of this country, $7\%$ of the inhabitants are infected with this disease. Among infected individuals, $20\%$ test negative. Among healthy individuals, $1\%$ test positive.
A person is chosen at random from the population. We denote:

• $M$ the event: ``the person is infected with the disease'';
• $T$ the event: ``the test is positive''.

1. Construct a probability tree modelling the proposed situation.
2. a. What is the probability that the person is infected with the disease and that their test is positive? b. Show that the probability that their test is positive is 0.0653.
3. It is known that the test of the chosen person is positive. What is the probability that they are infected? Give the result as an approximation to $10 ^ { - 2 }$ near.
4. Ten people are chosen at random from the population. The size of the population of this country allows us to treat this sample as a draw with replacement. Let $X$ be the random variable that counts the number of individuals with a positive test among the ten people. a. What is the probability distribution followed by $X$? Specify its parameters. b. Determine the probability that exactly two people have a positive test. Give the result as an approximation to $10 ^ { - 2 }$ near.
5. Determine the minimum number of people to test in this country so that the probability that at least one of these people has a positive test is greater than $99\%$.

Exercise 1 (7 points) Theme: probability The coyote is a wild animal close to the wolf, which lives in North America. In the state of Oklahoma, in the United States, $70\%$ of coyotes are affected by a disease called ehrlichiosis. There is a test that helps detect this disease. When this test is applied to a coyote, its result is either positive or negative, and we know that:

• If the coyote is sick, the test is positive in $97\%$ of cases.
• If the coyote is not sick, the test is negative in $95\%$ of cases.

Part A Veterinarians capture a coyote from Oklahoma at random and perform a test for ehrlichiosis on it. We consider the following events:

• $M$: ``the coyote is sick'';
• $T$: ``the coyote's test is positive''.

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

1. Copy and complete the probability tree below that models the situation.
2. Determine the probability that the coyote is sick and that its test is positive.
3. Prove that the probability of $T$ is equal to 0.694.
4. The ``positive predictive value of the test'' is called the probability that the coyote is actually sick given that its test is positive. Calculate the positive predictive value of the test. Round the result to the nearest thousandth.
5. a. By analogy with the previous question, propose a definition of the ``negative predictive value of the test'' and calculate this value rounding to the nearest thousandth. b. Compare the positive and negative predictive values of the test, and interpret.

Part B Recall that the probability that a randomly captured coyote has a positive test is 0.694.

1. When five coyotes are captured at random, this choice is treated as sampling with replacement. We denote $X$ the random variable that associates to a sample of five randomly captured coyotes the number of coyotes in this sample having a positive test. a. What is the probability distribution followed by $X$? Justify and specify its parameters. b. Calculate the probability that in a sample of five randomly captured coyotes, only one has a positive test. Round the result to the nearest hundredth. c. A veterinarian claims that there is more than a one in two chance that at least four out of five coyotes have a positive test: is this claim true? Justify your answer.
2. To test medications, veterinarians need to have a coyote with a positive test. How many coyotes must they capture so that the probability that at least one of them has a positive test is greater than 0.99?

According to the health authorities of a country, $7\%$ of the inhabitants are affected by a certain disease. In this country, a test is developed to detect this disease. This test has the following characteristics:

• For sick individuals, the test gives a negative result in $20\%$ of cases;
• For healthy individuals, the test gives a positive result in $1\%$ of cases.

A person is chosen at random from the population and tested. Consider the following events:

• $M$ ``the person is sick'';
• $T$ ``the test is positive''.

1. Calculate the probability of the event $M \cap T$. You may use a probability tree.
2. Prove that the probability that the test of the randomly chosen person is positive is $0.0653$.
3. In the context of disease screening, is it more relevant to know $P_M(T)$ or $P_T(M)$?
4. In this question, consider that the randomly chosen person had a positive test. What is the probability that they are sick? Round the result to $10^{-2}$ near.
5. People are chosen at random from the population. The size of the population of this country allows us to treat this sampling as drawing with replacement. Let $X$ be the random variable that gives the number of individuals with a positive test among 10 people. a. Specify the nature and parameters of the probability distribution followed by $X$. b. Determine the probability that exactly two people have a positive test. Round the result to $10^{-2}$ near.
6. Determine the minimum number of people to test in this country so that the probability that at least one of them has a positive test is greater than $99\%$.

Paratuberculosis is an infectious digestive disease that affects cows. It is caused by the presence of a bacterium in the cow's intestine.
A study is conducted in a region where $0.4\%$ of the cow population is infected.
There is a test that reveals the immune response of an organism infected by the bacterium. The result of this test can be either ``positive'' or ``negative''.
A cow is chosen at random from the region. Given the characteristics of the test, we know that:

• If the cow is affected by the infection, the probability that its test is positive is 0.992;
• If the cow is not affected by the infection, the probability that its test is negative is 0.984.

We denote by $I$ the event ``the cow is affected by the infection'' and $T$ the event ``the cow presents a positive test''. We denote by $\bar{I}$ the complementary event of $I$ and $\bar{T}$ the complementary event of $T$.
Part A

1. Reproduce and complete the weighted tree below modelling the situation.
2. a. Calculate the probability that the cow is not affected by the infection and that its test is negative. Give the result to $10^{-3}$ near. b. Show that the probability, to $10^{-3}$ near, that the cow presents a positive test is approximately equal to 0.020. c. The ``positive predictive value of the test'' is the probability that the cow is affected by the infection given that its test is positive. Calculate the positive predictive value of this test. Give the result to $10^{-3}$ near. d. The test gives incorrect information about the cow's state of health when the cow is not infected and presents a positive test result or when the cow is infected and presents a negative test result. Calculate the probability that this test gives incorrect information about the cow's state of health. Give a result to $10^{-3}$ near.

Part B

1. When a sample of 100 cows is chosen at random from the region, this choice is treated as a draw with replacement. Recall that, for a cow chosen at random from the region, the probability that the test is positive is equal to 0.02. We denote by $X$ the random variable that associates to a sample of 100 cows from the region chosen at random the number of cows presenting a positive test in this sample. a. What is the probability distribution followed by the random variable $X$? Justify the answer and specify the parameters of this distribution. b. Calculate the probability that in a sample of 100 cows, there are exactly 3 cows presenting a positive test. Give a result to $10^{-3}$ near. c. Calculate the probability that in a sample of 100 cows, there are at most 3 cows presenting a positive test. Give a result to $10^{-3}$ near.
2. We now choose a sample of $n$ cows from this region, $n$ being a non-zero natural integer. We admit that this choice can be treated as a draw with replacement. Determine the minimum value of $n$ so that the probability that there is, in the sample, at least one cow tested positive, is greater than or equal to 0.99.

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.
A test has been implemented: this allows to determine (even long after infection), whether or not a person has already been infected with COVID 19. If the test is positive, this means that the person has already been infected with COVID 19.
The sensitivity of a test is the probability that it is positive given that the person has been infected with the disease. The specificity of a test is the probability that the test is negative given that the person has not been infected with the disease.
The test manufacturer provides the following characteristics:

• Its sensitivity is 0.8.
• Its specificity is 0.99.

An individual is drawn and subjected to the test from the adult French population on May 11, 2020. Let $T$ be the event ``the test performed is positive''.

1. Complete the probability tree with the data from the statement.
2. Show that $p(T) = 0.05503$.
3. What is the probability that an individual has been infected given that their test is positive? Give an approximate value to $10^{-4}$ near of the result.

A test developed to detect Covid gives the correct diagnosis for $99\%$ of people with Covid. It also gives the correct diagnosis for $99\%$ of people without Covid. In a city $\frac{1}{1000}$ of the population has Covid.
If the probability is $x\%$, then your answer should be the integer closest to $x$. E.g., for probability $\frac{1}{3} = 33.33\ldots\%$, you should type 33 as your answer. For probability $\frac{2}{3}$ you should type 67 as your answer.
Suppose that a randomly selected person tested positive. What is the probability that this person has Covid? [2 points]

Shubhaangi thinks she may be allergic to Bengal gram and takes a test that is known to give the following results:

• For people who really do have the allergy, the test says ``Yes'' $90 \%$ of the time.
• For people who do not have the allergy, the test says ``Yes'' $15 \%$ of the time.

If $2 \%$ of the population has the allergy and Shubhaangi's test says ``Yes'', then the chances that Shubhaangi does really have the allergy are
(A) $1 / 9$
(B) $6 / 55$
(C) $1 / 11$
(D) cannot be determined from the given data.

In a group of 400 people, 160 are smokers and non-vegetarian; 100 are smokers and vegetarian and the remaining 140 are non-smokers and vegetarian. Their chances of getting a particular chest disorder are $35 \% , 20 \%$ and $10 \%$ respectively. A person is chosen from the group at random and is found to be suffering from the chest disorder. The probability that the selected person is a smoker and non-vegetarian is:
(1) $\frac { 14 } { 45 }$
(2) $\frac { 7 } { 45 }$
(3) $\frac { 8 } { 45 }$
(4) $\frac { 28 } { 45 }$

$25\%$ of the population are smokers. A smoker has 27 times more chances to develop lung cancer than a non-smoker. A person is diagnosed with lung cancer and the probability that this person is a smoker is $\frac { k } { 10 }$. Then the value of $k$ is $\_\_\_\_$.

It is known that 30\% of the population in a certain region is infected with a certain infectious disease. For a rapid screening test of the disease, there are two results: positive or negative. The test has an 80\% probability of identifying an infected person as positive and a 60\% probability of identifying an uninfected person as negative. To reduce the situation where the test incorrectly identifies an infected person as negative, experts recommend three consecutive tests. If $P$ is the probability that an infected person is among those who test negative in a single test, and $P'$ is the probability that an infected person is among those who test negative in all three consecutive tests, what is $\frac { P } { P' }$ closest to?
(1) 7
(2) 8
(3) 9
(4) 10
(5) 11