Thomas owns an MP3 player on which he has stored several thousand musical pieces. The set of musical pieces he owns is divided into three distinct genres according to the following distribution: 30\% classical music, 45\% variety, the rest being jazz. Thomas used two encoding qualities to store his musical pieces: high quality encoding and standard encoding. We know that:

• $\frac { 5 } { 6 }$ of the classical music pieces are encoded in high quality.
• $\frac { 5 } { 9 }$ of the variety pieces are encoded in standard quality.

We consider the following events: $C$ : ``The piece heard is a classical music piece''; $V :$ ``The piece heard is a variety piece''; $J$ : ``The piece heard is a jazz piece''; $H$ : ``The piece heard is encoded in high quality''; $S$ : ``The piece heard is encoded in standard quality''.

Part 1

Thomas decides to listen to a piece at random from all the pieces stored on his MP3 using the ``random play'' function. A probability tree may be helpful.

1. What is the probability that it is a classical music piece encoded in high quality?
2. We know that $P ( H ) = \frac { 13 } { 20 }$. a. Are the events $C$ and $H$ independent? b. Calculate $P ( J \cap H )$ and $P _ { J } ( H )$.

Part 2

During a long train journey, Thomas listens to, using the ``random play'' function of his MP3, 60 musical pieces.

1. Determine the asymptotic fluctuation interval at the 95\% threshold of the proportion of classical music pieces in a sample of size 60.
2. Thomas counted that he had listened to 12 classical music pieces during his journey. Can we think that the ``random play'' function of Thomas's MP3 player is defective?

Part 3

Consider the random variable $X$ which, to each song stored on the MP3 player, associates its duration expressed in seconds, and we establish that $X$ follows the normal distribution with mean 200 and standard deviation 20.
We listen to a musical piece at random.

1. Give an approximate value to $10 ^ { - 3 }$ of $P ( 180 \leqslant X \leqslant 220 )$.
2. Give an approximate value to $10 ^ { - 3 }$ of the probability that the piece heard lasts more than 4 minutes.

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

A student must go to his school each morning by 8:00 a.m. He takes the bicycle 7 days out of 10 and the bus the rest of the time. On days when he takes the bicycle, he arrives on time in $99.4\%$ of cases and when he takes the bus, he arrives late in $5\%$ of cases. A date is chosen at random during the school period and we denote by $V$ the event ``The student goes to school by bicycle'', $B$ the event ``the student goes to school by bus'' and $R$ the event ``The student arrives late at school''.

1. Translate the situation using a probability tree.
2. Determine the probability of the event $V \cap R$.
3. Prove that the probability of the event $R$ is 0.0192
4. On a given day, the student arrived late at school. What is the probability that he went there by bus?

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.

For each of the following five statements, indicate whether it is true or false and justify the answer. An unjustified answer is not taken into account. An absence of answer is not penalized.

1. Zoé goes to work on foot or by car. Where she lives, it rains one day out of four. When it rains, Zoé goes to work by car in $80\%$ of cases. When it does not rain, she goes to work on foot with a probability equal to 0.6.
   Statement $\mathbf{n^o 1}$: ``Zoé uses the car one day out of two.''
2. In the set $E$ of outcomes of a random experiment, we consider two events $A$ and $B$.
   Statement $\mathbf{n^o 2}$: ``If $A$ and $B$ are independent, then $A$ and $\bar{B}$ are also independent.''
3. We model the waiting time, expressed in minutes, at a counter, by a random variable $T$ that follows the exponential distribution with parameter 0.7.
   Statement $\mathbf{n^o 3}$: ``The probability that a customer waits at least five minutes at this counter is approximately 0.7.''
   Statement $\mathbf{n^o 4}$: ``The average waiting time at this counter is seven minutes.''
4. We know that $39\%$ of the French population has blood group A+. We want to know if this proportion is the same among blood donors. We survey 183 blood donors and among them, $34\%$ have blood group A+.
   Statement $\mathbf{n^o 5}$: ``We cannot reject, at the $5\%$ significance level, the hypothesis that the proportion of people with blood group A+ among blood donors is $39\%$ as in the general population.''

It is now assumed that, in the store:

• $80\%$ of the padlocks offered for sale are budget models, the others are high-end;
• $3\%$ of high-end padlocks are defective;
• $7\%$ of padlocks are defective.

A padlock is randomly selected from the store. We denote:

• $p$ the probability that a budget padlock is defective;
• $H$ the event: ``the selected padlock is high-end'';
• $D$ the event: ``the selected padlock is defective''.

1. Represent the situation using a probability tree.
2. Express $P(D)$ as a function of $p$. Deduce the value of the real number $p$.

Is the result obtained consistent with that of question A-2?
3. The selected padlock is in good condition. Determine the probability that it is a high-end padlock.

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?

A factory manufactures an electronic component. Two production lines are used. Production line A produces $40\%$ of the components and production line B produces the rest. Some of the manufactured components have a defect that prevents them from operating at the speed specified by the manufacturer. At the output of line A, $20\%$ of the components have this defect while at the output of line B, only $5\%$ do. A component manufactured in this factory is chosen at random. We denote: A the event ``the component comes from line A'', $B$ the event ``the component comes from line B'', S the event ``the component is defect-free''.

1. Show that the probability of event $S$ is $P(S) = 0.89$.
2. Given that the component has no defect, determine the probability that it comes from line A. The result should be given to the nearest $10^{-2}$.

Part B
This same entrepreneur decides to install anti-spam software. This software detects unwanted messages called spam (malicious messages, advertisements, etc.) and moves them to a file called the ``spam folder''. The manufacturer claims that $95\%$ of spam messages are moved. For his part, the entrepreneur knows that $60\%$ of the messages he receives are spam. After installing the software, he observes that $58.6\%$ of messages are moved to the spam folder. For a message chosen at random, we consider the following events:

• $D$: ``the message is moved'';
• $S$: ``the message is spam''.

1. Calculate $P ( S \cap D )$.
2. A message that is not spam is chosen at random. Show that the probability that it is moved equals 0.04.
3. A message that is not moved is chosen at random. What is the probability that this message is spam?
4. For the software chosen by the company, the manufacturer estimates that $2.7\%$ of messages moved to the spam folder are reliable messages. In order to test the software's effectiveness, the secretariat takes the trouble to count the number of reliable messages among the moved messages. It finds 13 reliable messages among the 231 messages moved during one week. Do these results call into question the manufacturer's claim?

Part B - Reaching an operator
If the waiting time before reaching an operator exceeds 5 minutes, the call automatically ends. Otherwise, the caller reaches an operator. We randomly choose a customer who calls the assistance line. We assume that the probability that the call comes from an Internet customer is 0.7. Furthermore, according to Part A, we take the following data:

• If the call comes from an Internet customer then the probability of reaching an operator is equal to 0.95.
• If the call comes from a mobile customer then the probability of reaching an operator is equal to 0.87.

1. Determine the probability that the customer reaches an operator.
2. A customer complains that their call ended after 5 minutes of waiting without reaching an operator. Is it more likely that this is an Internet customer or a mobile customer?

Machine A produces one third of the factory's sweets. The rest of production is ensured by machine B. When produced by machine B, the probability that a randomly selected sweet is deformed is equal to 0.02. In a quality control test, a sweet is randomly selected from the entire production. It is deformed.
What is the probability, rounded to the nearest hundredth, that it was produced by machine B?
Answer a: 0.02 Answer b: 0.67 Answer c: 0.44 Answer d: 0.01

The municipality of a large city has a stock of DVDs that it offers for rental to users of the various media libraries in this city. Among the DVDs removed, some are defective, others are not. 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.
A DVD is chosen at random from the municipality's stock. Consider the following events:

• $D$: ``the DVD is defective'';
• $R$: ``the DVD is removed from stock''.

We denote by $\bar{D}$ and $\bar{R}$ the complementary events of events $D$ and $R$ respectively.

1. Prove that the probability of event $R$ is 0.134.
2. A charitable association contacts the municipality with the aim of recovering all DVDs that are removed from stock. A city official then claims that among these removed DVDs, more than half are composed of defective DVDs. Is this claim true?

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. We admit that $3\%$ of the sugar from farm $U$ is extra fine and that $5\%$ of the sugar from farm V is extra fine. A packet of sugar is randomly selected from the company's production and we consider the following events:

• $U$: ``The packet contains sugar from farm U'';
• $V$: ``The packet contains sugar from farm V'';
• $E$: ``The packet bears the label `extra fine' ''.

1. In this question, we admit that the company manufactures $30\%$ of its packets with sugar from farm U and the others with sugar from farm V, without mixing sugars from the two farms. a. What is the probability that the selected packet bears the label ``extra fine''? b. Given that a packet bears the label ``extra fine'', what is the probability that the sugar it contains comes from farm U?
2. The company wishes to modify its supply from the two farms so that among the packets bearing the label ``extra fine'', $30\%$ of them contain sugar from farm U. How should it supply itself from farms U and V? Any working will be valued in this question.

Part A

Louise drives to work with her car. Her colleague Zoé does not own a car. Each morning, Louise therefore offers to give Zoé a ride. Whatever Zoé's answer, Louise offers to drive her back in the evening. We consider a given day. We have the following information:

• the probability that Louise drives Zoé in the morning is 0.55;
• if Louise drove Zoé in the morning, the probability that she drives her back in the evening is 0.7;
• if Louise did not drive Zoé in the morning, the probability that she drives her back in the evening is 0.24.

We denote $M$ and $S$ the following events:

• $M$: ``Louise drives Zoé in the morning'';
• S: ``Louise drives Zoé back in the evening''.

1. Construct a probability tree representing the situation.
2. Calculate $P ( M \cap S )$. Translate this result with a sentence.
3. Prove that the probability of event S is equal to 0.493.
4. We know that Louise drove Zoé back in the evening. What is the probability that Louise drove her in the morning?

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.

In a statistics school, after reviewing candidate files, recruitment is done in two ways:

• $10\%$ of candidates are selected based on their file. These candidates must then take an oral examination after which $60\%$ of them are finally admitted to the school.
• Candidates who were not selected based on their file take a written examination after which $20\%$ of them are admitted to the school.

Part 1
A candidate for this recruitment competition is chosen at random. We denote:

• $D$ the event ``the candidate was selected based on their file'';
• $A$ the event ``the candidate was admitted to the school'';
• $\bar{D}$ and $\bar{A}$ the complementary events of events $D$ and $A$ respectively.

1. Represent the situation with a probability tree.
2. Calculate the probability that the candidate is selected based on their file and admitted to the school.
3. Show that the probability of event $A$ is equal to 0.24.
4. A candidate admitted to the school is chosen at random. What is the probability that their file was not selected?

Part 2

1. We assume that the probability for a candidate to be admitted to the school is equal to 0.24.
   We consider a sample of seven candidates chosen at random, treating this choice as a random draw with replacement. We denote by $X$ the random variable counting the candidates admitted to the school among the seven drawn. a. We assume that the random variable $X$ follows a binomial distribution. What are the parameters of this distribution? b. Calculate the probability that only one of the seven candidates drawn is admitted to the school. Give an answer rounded to the nearest hundredth. c. Calculate the probability that at least two of the seven candidates drawn are admitted to this school. Give an answer rounded to the nearest hundredth.
2. A secondary school presents $n$ candidates for recruitment in this school, where $n$ is a non-zero natural number. We assume that the probability for any candidate from the secondary school to be admitted to the school is equal to 0.24 and that the results of the candidates are independent of each other. a. Give the expression, as a function of $n$, of the probability that no candidate from this secondary school is admitted to the school. b. From what value of the integer $n$ is the probability that at least one student from this secondary school is admitted to the school greater than or equal to 0.99?

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.

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 — Theme: Probability Results should be rounded if necessary to $10^{-4}$
A statistical study conducted in a company provides the following information:

• $48\%$ of employees are women. Among them, $16.5\%$ hold a managerial position;
• $52\%$ of employees are men. Among them, $21.5\%$ hold a managerial position.

A person is chosen at random from among the employees. The following events are considered:

• $F$: ``the chosen person is a woman'';
• $C$: ``the chosen person holds a managerial position''.

1. Represent the situation with a probability tree.
2. Calculate the probability that the chosen person is a woman who holds a managerial position.
3. a. Prove that the probability that the chosen person holds a managerial position is equal to 0.191. b. Are the events $F$ and $C$ independent? Justify.
4. Calculate the probability of $F$ given $C$, denoted $P_{C}(F)$. Interpret the result in the context of the exercise.
5. A random sample of 15 employees is chosen. The large number of employees in the company allows this choice to be treated as sampling with replacement. Let $X$ be the random variable giving the number of managers in the sample of 15 employees. Recall that the probability that a randomly chosen employee is a manager is equal to 0.191. a. Justify that $X$ follows a binomial distribution and specify its parameters. b. Calculate the probability that the sample contains at most 1 manager. c. Determine the expected value of the random variable $X$.
6. Let $n$ be a natural number. In this question, consider a sample of $n$ employees. What must be the minimum value of $n$ so that the probability that there is at least one manager in the sample is greater than or equal to 0.99?

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?

Exercise 1 (7 points) -- Probabilities
Among sore throats, one quarter requires taking antibiotics, the others do not. In order to avoid unnecessarily prescribing antibiotics, doctors have a diagnostic test with the following characteristics:

• when the sore throat requires taking antibiotics, the test is positive in $90\%$ of cases;
• when the sore throat does not require taking antibiotics, the test is negative in $95\%$ of cases.

The probabilities requested in the rest of the exercise will be rounded to $10^{-4}$ if necessary.
Part 1
A patient with a sore throat who has undergone the test is chosen at random. Consider the following events:

• $A$: ``the patient has a sore throat requiring taking antibiotics'';
• $T$: ``the test is positive'';
• $\bar{A}$ and $\bar{T}$ are respectively the complementary events of $A$ and $T$.

1. Calculate $P(A \cap T)$. You may use a probability tree.
2. Prove that $P(T) = 0.2625$.
3. A patient with a positive test is chosen. Calculate the probability that they have a sore throat requiring taking antibiotics.
4. a. Among the following events, determine which correspond to an incorrect test result: $A \cap T,\ \bar{A} \cap T,\ A \cap \bar{T},\ \bar{A} \cap \bar{T}$. b. Define the event $E$: ``the test gives an incorrect result''. Prove that $P(E) = 0.0625$.

Part 2
A sample of $n$ patients who have been tested is selected at random. We assume that this sample selection can be treated as sampling with replacement. Let $X$ be the random variable giving the number of patients in this sample with an incorrect test result.

1. Suppose that $n = 50$. a. Justify that the random variable $X$ follows a binomial distribution $\mathscr{B}(n, p)$ with parameters $n = 50$ and $p = 0.0625$. b. Calculate $P(X = 7)$. c. Calculate the probability that there is at least one patient in the sample whose test is incorrect.
2. What is the minimum sample size needed so that $P(X \geqslant 10)$ is greater than $0.95$?

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 $99\%$.

Customs authorities are interested in imports of headphones bearing the logo of a certain brand. Customs seizures allow them to estimate that:

• $20 \%$ of headphones bearing this brand's logo are counterfeits;
• $2 \%$ of non-counterfeit headphones have a design defect;
• $10 \%$ of counterfeit headphones have a design defect.

The fraud agency randomly orders a headphone displaying the brand's logo from an internet site. Consider the following events:

• C: ``the headphone is counterfeit'';
• $D$: ``the headphone has a design defect'';
• $\bar { C }$ and $\bar { D }$ denote respectively the complementary events of $C$ and $D$.

Throughout the exercise, probabilities will be rounded to $10 ^ { - 3 }$ if necessary.

Part 1

1. Calculate $P ( C \cap D )$. You may use a probability tree.
2. Prove that $P ( D ) = 0,036$.
3. The headphone has a defect. What is the probability that it is counterfeit?

Part 2

We order $n$ headphones bearing this brand's logo. We treat this experiment as a random draw with replacement. Let $X$ be the random variable giving the number of headphones with a design defect in this batch.

1. In this question, $n = 35$. a. Justify that $X$ follows a binomial distribution $\mathscr { B } ( n , p )$ where $n = 35$ and $p = 0,036$. b. Calculate the probability that among the ordered headphones, exactly one has a design defect. c. Calculate $P ( X \leqslant 1 )$.
2. In this question, $n$ is not fixed. What is the minimum number of headphones to order so that the probability that at least one headphone has a defect is greater than 0.99?

Part 1

Julien must take the plane; he planned to take the bus to get to the airport. If he takes the 8 o'clock bus, he is sure to be at the airport in time for his flight. On the other hand, the next bus would not allow him to arrive at the airport in time. Julien left late from his apartment and the probability that he misses his bus is 0.8. If he misses his bus, he goes to the airport by taking a private car company; he then has a probability of 0.5 of being on time at the airport. We denote:

• $B$ the event: ``Julien manages to take his bus'';
• $V$ the event: ``Julien is on time at the airport for his flight''.

1. Give the value of $P _ { B } ( V )$.
2. Represent the situation with a probability tree.
3. Show that $P ( V ) = 0.6$.
4. If Julien is on time at the airport for his flight, what is the probability that he arrived at the airport by bus? Justify.

Part 2

Airlines sell more tickets than there are seats on planes because some passengers do not show up for boarding on the flight they have booked. This practice is called overbooking. Based on statistics from previous flights, the airline estimates that each passenger has a 5\% chance of not showing up for boarding. Consider a flight on a plane with 200 seats for which 206 tickets have been sold. We assume that the presence at boarding of each passenger is independent of other passengers and we call $X$ the random variable that counts the number of passengers showing up for boarding.

1. Justify that $X$ follows a binomial distribution and specify its parameters.
2. On average, how many passengers will show up for boarding?
3. Calculate the probability that 201 passengers show up for boarding. The result should be rounded to $10 ^ { - 3 }$.
4. Calculate $P ( X \leqslant 200 )$, the result should be rounded to $10 ^ { - 3 }$. Interpret this result in the context of the exercise.
5. The airline sells each ticket for 250 euros.

If more than 200 passengers show up for boarding, the airline must refund the plane ticket and pay a penalty of 600 euros to each affected passenger. We call: $Y$ the random variable equal to the number of passengers who cannot board despite having purchased a ticket; $C$ the random variable that totals the revenue of the airline on this flight.
We admit that $Y$ follows the probability distribution given by the following table:

$y _ { i }$	0	1	2	3	4	5	6
$P \left( Y = y _ { i } \right)$	0,94775	0,03063	0,01441	0,00539	0,00151	0,00028

a. Complete the probability distribution given above by calculating $P ( Y = 6 )$. b. Justify that: $C = 51500 - 850 Y$. c. Give the probability distribution of the random variable $C$ in the form of a table. Calculate the expected value of the random variable $C$ to the nearest euro. d. Compare the revenue obtained by selling exactly 200 tickets and the average revenue obtained by practicing overbooking.