A chain of stores wishes to build customer loyalty by offering gift vouchers to its privileged customers. Each of them receives a gift voucher of green or red colour on which an amount is written. Gift vouchers are distributed so as to have, in each store, one quarter of red vouchers and three quarters of green vouchers. Green gift vouchers take the value of 30 euros with a probability equal to 0.067 or values between 0 and 15 euros with unspecified probabilities here. Similarly, red gift vouchers take the values 30 or 100 euros with probabilities respectively equal to 0.015 and 0.010 or values between 10 and 20 euros with unspecified probabilities here.
Calculate the probability of having a gift voucher with a value greater than or equal to 30 euros knowing that it is red.

Exercise 1

4 points

Common to all candidates

Parts $\mathrm { A } , \mathrm { B }$ and C can be treated independently. Throughout the exercise, results should be rounded, if necessary, to the nearest thousandth.

Part A

A merchant receives the results of a market study on consumer habits in France. According to this study:

• $54 \%$ of consumers prefer products of French manufacture;
• $65 \%$ of consumers regularly buy products from organic agriculture, and among them $72 \%$ prefer products of French manufacture.

A consumer is chosen at random. The following events are considered:

• B: ``a consumer regularly buys products from organic agriculture'';
• $F$ : ``a consumer prefers products of French manufacture''.

We denote $P ( A )$ the probability of event $A$ and $P _ { C } ( A )$ the probability of $A$ given $C$.

1. Justify that $P ( \bar { B } \cap F ) = 0.072$.
2. Calculate $P _ { F } ( \bar { B } )$.
3. A consumer is chosen who does not regularly buy products from organic agriculture. What is the probability that he prefers products of French manufacture?

Part B

The merchant is interested in the quantity in kilograms of organic flour sold each month at retail in his store. This quantity is modeled by a random variable $X$ which follows the normal distribution with mean $\mu = 90$ and standard deviation $\sigma = 2$.

1. At the beginning of each month, the merchant ensures he has 95 kg in stock.

What is the probability that he cannot meet customer demand during the month?
2. Determine an approximate value to the nearest hundredth of the real number $a$ such that $P ( X < a ) = 0.02$.
Interpret the result in the context of the exercise.

Part C

In this market study, it is specified that $46.8 \%$ of consumers in France prefer local products. The merchant observes that among his 2500 customers, 1025 regularly buy local products. Is his customer base representative of consumers in France?

A technician controls the machines equipping a large company. All these machines are identical. We know that:

• $20\%$ of machines are under warranty;
• $0.2\%$ of machines are both defective and under warranty;
• $8.2\%$ of machines are defective.

The technician tests a machine at random. We consider the following events:

• G: ``the machine is under warranty'';
• $D$: ``the machine is defective'';
• $\bar{G}$ and $\bar{D}$ denote respectively the complementary events of $G$ and $D$.

The probability $p_{G}(D)$ of event $D$ given that $G$ is realized is equal to: a. 0.002 b. 0.01 c. 0.024 d. 0.2

Here is the distribution of the main blood groups of the inhabitants of France. $A+, O+, B+, A-, O-, AB+, B-$ and $AB-$ are the different blood groups combined with the Rh factor.
A random experiment consists of choosing a person at random from the French population and determining their blood group and Rh factor.
We adopt notations of the type: $A+$ is the event ``the person has blood group A and Rh factor $+$'' $A-$ is the event ``the person has blood group A and Rh factor $-$'' $A$ is the event ``the person has blood group A''
Parts 1 and 2 are independent.
Part 1
We denote $Rh+$ the event ``The person has positive Rh factor''.

1. Justify that the probability that the chosen person has positive Rh factor is equal to 0.849.
2. Demonstrate using the data from the problem statement that $P_{Rh_+}(A) = 0.450$ to 0.001 near.
3. A person remembers that their blood group is AB but has forgotten their Rh factor. What is the probability that their Rh factor is negative? Round the result to 0.001 near.

Part 2
In this part, results will be rounded to 0.001 near.
A universal blood donor is a person with blood group O and negative Rh factor. Recall that $6.5\%$ of the French population has blood group $O-$.

1. We consider 50 people chosen at random from the French population and we denote $X$ the random variable that counts the number of universal donors. a. Determine the probability that 8 people are universal donors. Justify your answer. b. Consider the function below named \texttt{proba} with argument \texttt{k} written in Python language. \begin{verbatim} def proba(k) : p = 0 for i in range(k+1) : p = p + binomiale(i,50,0.065) return p \end{verbatim} This function uses the binomial function with arguments $i, n$ and $p$, created for this purpose, which returns the value of the probability $P(X=i)$ in the case where $X$ follows a binomial distribution with parameters $n$ and $p$. Determine the numerical value returned by the \texttt{proba} function when you enter \texttt{proba(8)} in the Python console. Interpret this result in the context of the exercise.
2. What is the minimum number of people to choose at random from the French population so that the probability that at least one of the chosen people is a universal donor is greater than 0.999.

In a school with 1200 students, a survey was conducted on their knowledge of two foreign languages, English and Spanish.
In this survey, it was found that 600 students speak English, 500 speak Spanish, and 300 do not speak either of these languages.
If a student from this school is chosen at random and it is known that he does not speak English, what is the probability that this student speaks Spanish?
(A) $\frac{1}{2}$ (B) $\frac{5}{8}$ (C) $\frac{1}{4}$ (D) $\frac{5}{6}$ (E) $\frac{5}{14}$

A certain class consists of 18 male students and 16 female students. All students in this class take a class in either Chinese or Japanese, but not both. Among the male students, 12 take Chinese class, and among the female students, 7 take Japanese class. When a student selected from this class is taking Chinese class, what is the probability that this student is female? [3 points]
(1) $\frac { 1 } { 7 }$
(2) $\frac { 2 } { 7 }$
(3) $\frac { 3 } { 7 }$
(4) $\frac { 4 } { 7 }$
(5) $\frac { 5 } { 7 }$

For two events $A , B$, $$\mathrm { P } ( A ) = \frac { 1 } { 4 } , \quad \mathrm { P } ( B ) = \frac { 2 } { 3 } , \quad A \subset B$$ What is the value of $\mathrm { P } ( A \mid B )$? [3 points]
(1) $\frac { 1 } { 8 }$
(2) $\frac { 1 } { 4 }$
(3) $\frac { 3 } { 8 }$
(4) $\frac { 1 } { 2 }$
(5) $\frac { 5 } { 8 }$

For two events $A$ and $B$, $\mathrm { P } ( A ) = \frac { 1 } { 2 } , \mathrm { P } \left( B ^ { C } \right) = \frac { 2 } { 3 }$, and $\mathrm { P } ( B \mid A ) = \frac { 1 } { 6 }$. What is the value of $\mathrm { P } \left( A ^ { C } \mid B \right)$? (Here, $A ^ { C }$ is the complement of $A$.) [3 points]
(1) $\frac { 1 } { 2 }$
(2) $\frac { 7 } { 12 }$
(3) $\frac { 2 } { 3 }$
(4) $\frac { 3 } { 4 }$
(5) $\frac { 5 } { 6 }$

For two events $A, B$, $$\mathrm{P}(A \cap B) = \frac{1}{8}, \quad \mathrm{P}\left(B^{C} \mid A\right) = 2\mathrm{P}(B \mid A)$$ what is the value of $\mathrm{P}(A)$? (Here, $B^{C}$ is the complement of $B$.) [3 points]
(1) $\frac{5}{12}$
(2) $\frac{3}{8}$
(3) $\frac{1}{3}$
(4) $\frac{7}{24}$
(5) $\frac{1}{4}$

Among 50 members of a marathon club who participated in a certain marathon, the number of members who completed the marathon and the number who withdrew are as follows. (Unit: persons)

Category	Male	Female
Completed	27	9
Withdrew	8	6

When one member is randomly selected from the participants and is found to be female, the probability that this member completed the marathon is $p$. Find the value of $100 p$. [3 points]

For two events $A$ and $B$, $$\mathrm { P } ( A ) = \frac { 1 } { 3 } , \quad \mathrm { P } ( A \cap B ) = \frac { 1 } { 8 }$$ what is the value of $\mathrm { P } \left( B ^ { C } \mid A \right)$? (Here, $B ^ { C }$ is the complement of $B$.) [4 points]
(1) $\frac { 11 } { 24 }$
(2) $\frac { 1 } { 2 }$
(3) $\frac { 13 } { 24 }$
(4) $\frac { 7 } { 12 }$
(5) $\frac { 5 } { 8 }$

For two events $A , B$, $$\mathrm { P } ( A ) = \frac { 2 } { 5 } , \quad \mathrm { P } ( B \mid A ) = \frac { 5 } { 6 }$$ what is the value of $\mathrm { P } ( A \cap B )$? [3 points]
(1) $\frac { 1 } { 3 }$
(2) $\frac { 4 } { 15 }$
(3) $\frac { 1 } { 5 }$
(4) $\frac { 2 } { 15 }$
(5) $\frac { 1 } { 15 }$

A survey was conducted on 500 students at a high school regarding their desire to visit regions A and B for cultural exploration. The results are as follows. (Unit: students)

Region B	Wish	Do not wish	Total
Wish	140	310	450
Do not wish	40	10	50
Total	180	320	500

When one student is randomly selected from this high school and is found to wish to visit region A, what is the probability that this student also wishes to visit region B? [3 points]
(1) $\frac { 19 } { 45 }$
(2) $\frac { 23 } { 45 }$
(3) $\frac { 3 } { 5 }$
(4) $\frac { 31 } { 45 }$
(5) $\frac { 7 } { 9 }$

For two events $A , B$, $A$ and $B ^ { C }$ are mutually exclusive events, and $$\mathrm { P } ( A ) = \frac { 1 } { 3 } , \mathrm { P } \left( A ^ { C } \cap B \right) = \frac { 1 } { 6 }$$ What is the value of $\mathrm { P } ( B )$? (Here, $A ^ { C }$ is the complement of $A$.) [3 points]
(1) $\frac { 5 } { 12 }$
(2) $\frac { 1 } { 2 }$
(3) $\frac { 7 } { 12 }$
(4) $\frac { 2 } { 3 }$
(5) $\frac { 3 } { 4 }$

A survey was conducted on 200 students at a school regarding their preferences for experiential activities. The students who participated in this survey chose one of cultural experience or ecological research, and the number of students who chose each activity is as follows.

Classification	Cultural Experience	Ecological Research	Total
Male Students	40	60	100
Female Students	50	50	100
Total	90	110	200

When one student is randomly selected from the 200 students who participated in this survey and is a student who chose ecological research, what is the probability that this student is a female student? [3 points]
(1) $\frac { 5 } { 11 }$
(2) $\frac { 1 } { 2 }$
(3) $\frac { 6 } { 11 }$
(4) $\frac { 5 } { 9 }$
(5) $\frac { 3 } { 5 }$

For two events $A$ and $B$, $$\mathrm { P } ( B \mid A ) = \frac { 1 } { 4 } , \quad \mathrm { P } ( A \mid B ) = \frac { 1 } { 3 } , \quad \mathrm { P } ( A ) + \mathrm { P } ( B ) = \frac { 7 } { 10 }$$ what is the value of $\mathrm { P } ( A \cap B )$? [3 points]
(1) $\frac { 1 } { 7 }$
(2) $\frac { 1 } { 8 }$
(3) $\frac { 1 } { 9 }$
(4) $\frac { 1 } { 10 }$
(5) $\frac { 1 } { 11 }$

For two events $A$ and $B$, $$\mathrm{P}(A \mid B) = \mathrm{P}(A) = \frac{1}{2}, \quad \mathrm{P}(A \cap B) = \frac{1}{5}$$ What is the value of $\mathrm{P}(A \cup B)$? [3 points]
(1) $\frac{1}{2}$
(2) $\frac{3}{5}$
(3) $\frac{7}{10}$
(4) $\frac{4}{5}$
(5) $\frac{9}{10}$

For two events $A , B$, $$\mathrm { P } ( A ) = \frac { 2 } { 5 } , \quad \mathrm { P } ( B \mid A ) = \frac { 1 } { 4 } , \quad \mathrm { P } ( A \cup B ) = 1$$ What is the value of $\mathrm { P } ( B )$? [3 points]
(1) $\frac { 7 } { 10 }$
(2) $\frac { 3 } { 4 }$
(3) $\frac { 4 } { 5 }$
(4) $\frac { 17 } { 20 }$
(5) $\frac { 9 } { 10 }$

50 people registered for the soccer club, 60 people registered for the table tennis club, and 70 people registered for either the soccer or table tennis club. If a person is known to have registered for the soccer club, the probability that they also registered for the table tennis club is
A. $0.8$
B. $0.4$
C. $0.2$
D. $0.1$

Let $X$ and $Y$ be two events such that $P(X) = \frac{1}{3}$, $P(X \mid Y) = \frac{1}{2}$ and $P(Y \mid X) = \frac{2}{5}$. Then
[A] $P(Y) = \frac{4}{15}$
[B] $P(X' \mid Y) = \frac{1}{2}$
[C] $P(X \cap Y) = \frac{1}{5}$
[D] $P(X \cup Y) = \frac{2}{5}$

Three students $S _ { 1 } , S _ { 2 }$, and $S _ { 3 }$ are given a problem to solve. Consider the following events: $U$ : At least one of $S _ { 1 } , S _ { 2 }$, and $S _ { 3 }$ can solve the problem, $V : S _ { 1 }$ can solve the problem, given that neither $S _ { 2 }$ nor $S _ { 3 }$ can solve the problem, $W : S _ { 2 }$ can solve the problem and $S _ { 3 }$ cannot solve the problem, T: $S _ { 3 }$ can solve the problem.
For any event $E$, let $P ( E )$ denote the probability of $E$. If
$$P ( U ) = \frac { 1 } { 2 } , \quad P ( V ) = \frac { 1 } { 10 } , \quad \text { and } \quad P ( W ) = \frac { 1 } { 12 }$$
then $P ( T )$ is equal to

(A)

$\frac { 13 } { 36 }$

(B)

$\frac { 1 } { 3 }$

(C)

$\frac { 19 } { 60 }$

(D)

$\frac { 1 } { 4 }$

Let $A$ and $E$ be any two events with positive probabilities Statement I: $P ( E / A ) \geq P ( A / E ) P ( E )$. Statement II: $P ( A / E ) \geq P ( A \cap E )$.
(1) Both the statements are false
(2) Both the statements are true
(3) Statement-I is false, Statement-II is true
(4) Statement - I is true, Statement - II is false

Let $A$ and $B$ be two non-null events such that $A \subset B$. Then, which of the following statements is always correct?
(1) $P(A \mid B) \geq P(A)$
(2) $P(A \mid B) = P(B) - P(A)$
(3) $P(A \mid B) \leq P(A)$
(4) $P(A \mid B) = 1$

Let $A$ and $B$ be two events such that $P ( B \mid A ) = \frac { 2 } { 5 } , P ( A \mid B ) = \frac { 1 } { 7 }$ and $P ( A \cap B ) = \frac { 1 } { 9 }$. Consider $( S1 )\; P \left( A ^ { \prime } \cup B \right) = \frac { 5 } { 6 }$, $( S2 )\; P \left( A ^ { \prime } \cap B ^ { \prime } \right) = \frac { 1 } { 18 }$. Then
(1) Both $(S1)$ and $(S2)$ are true
(2) Both $(S1)$ and $(S2)$ are false
(3) Only $(S1)$ is true
(4) Only $(S2)$ is true

Let Ajay will not appear in JEE exam with probability $p = \frac{2}{7}$, while both Ajay and Vijay will appear in the exam with probability $q = \frac{1}{5}$. Then the probability, that Ajay will appear in the exam and Vijay will not appear is:
(1) $\frac{9}{35}$
(2) $\frac{18}{35}$
(3) $\frac{24}{35}$
(4) $\frac{3}{35}$