Chikungunya is a viral disease transmitted from one human to another by bites of infected female mosquitoes. A test has been developed for the screening of this virus. The laboratory manufacturing this test provides the following characteristics:

• the probability that a person affected by the virus has a positive test is 0.98;
• the probability that a person not affected by the virus has a positive test is 0.01.

An individual is chosen at random from a target population. We call:

• $M$ the event: "The chosen individual is affected by chikungunya"
• $T$ the event: "The test of the chosen individual is positive"

We denote by $\bar{M}$ (respectively $\bar{T}$) the opposite event of the event $M$ (respectively $T$). We denote by $p$ $(0 \leqslant p \leqslant 1)$ the proportion of people affected by the disease in the target population.

1. a. Copy and complete the probability tree. b. Express $P(M \cap T)$, $P(\bar{M} \cap T)$ then $P(T)$ as a function of $p$.
2. a. Prove that the probability of $M$ given $T$ is given by the function $f$ defined on $[0;1]$ by: $$f(p) = \frac{98p}{97p + 1}$$ b. Study the variations of the function $f$.
3. We consider that the test is reliable when the probability that a person with a positive test is actually affected by chikungunya is greater than 0.95. Using the results of question 2., from what proportion $p$ of sick people in the population is the test reliable?

4 points Parts $\boldsymbol{A}$ and $\boldsymbol{B}$ are independent. The requested probabilities will be given to $10^{-3}$ near. To help detect certain allergies, a blood test can be performed whose result is either positive or negative. In a population, this test gives the following results:

• If an individual is allergic, the test is positive in $97\%$ of cases;
• If an individual is not allergic, the test is negative in $95{,}7\%$ of cases.

Furthermore, $20\%$ of individuals in the concerned population have a positive test. We randomly choose an individual from the population, and we denote:

• $A$ the event ``the individual is allergic'';
• $T$ the event ``the individual has a positive test''.

We denote by $\bar{A}$ and $\bar{T}$ the complementary events of $A$ and $T$. We also call $x$ the probability of event $A$: $x = p(A)$.
Part A

1. Reproduce and complete the tree describing the situation, indicating on each branch the corresponding probability.
2. a. Prove the equality: $p(T) = 0{,}927x + 0{,}043$. b. Deduce the probability that the chosen individual is allergic.
3. Justify by a calculation the following statement: ``If the test of an individual chosen at random is positive, there is more than $80\%$ chance that this individual is allergic''.

Part B
A survey on allergies is conducted in a city by interviewing 150 randomly chosen residents, and we assume that this choice amounts to successive independent draws with replacement. We know that the probability that a randomly chosen resident in this city is allergic is equal to $0{,}08$. We denote by $X$ the random variable that associates to a sample of 150 randomly chosen residents the number of allergic people in this sample.

1. What is the probability distribution followed by the random variable $X$? Specify its parameters.
2. Determine the probability that exactly 20 people among the 150 interviewed are allergic.
3. Determine the probability that at least $10\%$ of the people among the 150 interviewed are allergic.

The function $f$ is defined on the interval $[ 0 ; 1 ]$ by: $$f ( x ) = \frac { 0,96 x } { 0,93 x + 0,03 }$$
The fight against doping involves carrying out anti-doping tests which aim to determine whether an athlete has used prohibited substances. During a competition bringing together 1000 athletes, a medical team tests all competitors. We propose to study the reliability of this test.
Let $x$ denote the real number between 0 and 1 which represents the proportion of doped athletes. During the development of this test, it was possible to determine that:

• the probability that an athlete is declared positive given that they are doped is equal to 0.96;
• the probability that an athlete is declared positive given that they are not doped is equal to 0.03.

We denote:

• D the event: ``the athlete is doped''.
• $T$ the event: ``the test is positive''.

1. Copy and complete the probability tree.
2. Determine, as a function of $x$, the probability that an athlete is doped and has a positive test.
3. Prove that the probability of event $T$ is equal to $0,93 x + 0,03$.
4. For this question only, assume that there are 50 doped athletes among the 1000 tested. Prove that the probability that an athlete is doped given that their test is positive is equal to $f ( 0,05 )$. Give an approximate value rounded to the nearest hundredth.
5. The positive predictive value of a test is called the probability that the athlete is truly doped when the test result is positive.
   1. [a.] Determine from which value of $x$ the positive predictive value of the test studied will be greater than or equal to 0.9. Round the result to the nearest hundredth.
   2. [b.] A competition official decides to no longer test all athletes, but to target the most successful athletes who are assumed to be more frequently doped. What is the consequence of this decision on the positive predictive value of the test? Argue using a result from Part A.

All probabilities, unless otherwise indicated, will be rounded to $10^{-3}$ in this exercise.
A test was developed for the detection of the chikungunya virus. The laboratory manufacturing this test provides the following characteristics:

• the probability that an individual affected by the virus has a positive test is 0.999;
• the probability that an individual not affected by the virus has a positive test is 0.005.

An individual is chosen at random from this population. We call:

• $M$ the event: ``the chosen individual is affected by chikungunya''.
• $T$ the event: ``the test of the chosen individual is positive''.

The test is considered reliable when the probability that an individual with a positive test is affected by the virus is greater than 0.95.

Part A: Study of an example

1. Give the probabilities $P_{M}(T)$ and $P_{\bar{M}}(T)$. ``In March 2005, the epidemic spread rapidly on the island of Réunion, with a major outbreak between late April and early June followed by persistence of viral transmission during the austral winter. In total, 270,000 people were infected out of a total population of 750,000 individuals''. At the end of 2005, the laboratory conducted a mass screening test of the population of the island of Réunion. In this part, the target population is therefore the population of the island of Réunion.
2. Give the exact value of $P(M)$.
3. Copy and complete the weighted tree.
4. Calculate the probability that an individual is affected by the virus and has a positive test.
5. Calculate the probability that an individual has a positive test.
6. Calculate the probability that an individual with a positive test is affected by the virus.
7. Can we estimate that this test is reliable? Argue.

Part B: Screening on a target population

In this part, we denote by $p$ the proportion of people affected by chikungunya virus in a target population. We seek here to test the reliability of this laboratory's test as a function of $p$.

1. Copy, adapting it, the weighted tree from question A3 taking into account the new data.
2. Express the probability $P(T)$ as a function of $p$.
3. Show that $P_{T}(M) = \frac{999p}{994p + 5}$.
4. For which values of $p$ can we consider that this test is reliable?

Part C: Study on a sample

During the epidemic, we assume that the probability of being affected by chikungunya on the island of Réunion is 0.36. We consider a sample of $n$ individuals chosen at random, assimilating this choice to a random draw with replacement. We denote by $X$ the random variable counting the number of infected individuals in this sample among the $n$ drawn. We assume that $X$ follows a binomial distribution with parameters $n$ and $p = 0.36$. Determine from how many individuals $n$ the probability of the event ``at least one of the $n$ inhabitants of this sample is affected by the virus'' is greater than 0.99. Explain the approach.

A research team uses a certain rapid test reagent to understand the proportion of organisms in a protected area whose body toxin accumulation exceeds the standard due to environmental pollution. The test result of this reagent shows only two colors: red and yellow. Based on past experience, it is known that: if body toxin accumulation exceeds the standard, after testing with this reagent, $75\%$ shows red; if body toxin accumulation does not exceed the standard, after testing with this reagent, $95\%$ shows yellow. It is known that for a certain type of organism in this protected area, $7.8\%$ of the test results show red. Assuming the actual proportion of this type of organism with body toxin accumulation exceeding the standard is $p\%$ , select the correct option.
(1) $1 \leq p < 3$
(2) $3 \leq p < 5$
(3) $5 \leq p < 7$
(4) $7 \leq p < 9$
(5) $9 \leq p < 11$