With three identical valves $V_1, V_2$ and $V_3$, we manufacture a hydraulic circuit. The circuit is operational if $V_1$ is operational or if $V_2$ and $V_3$ are simultaneously operational.
We treat as a random experiment the fact that each valve is or is not operational after 6000 hours. We denote:

• $F_1$ the event: ``valve $V_1$ is operational after 6000 hours''.
• $F_2$ the event: ``valve $V_2$ is operational after 6000 hours''.
• $F_3$ the event: ``valve $V_3$ is operational after 6000 hours''.
• $E$: the event: ``the circuit is operational after 6000 hours''.

We assume that the events $F_1, F_2$ and $F_3$ are pairwise independent and each have probability equal to 0.3.

1. The probability tree shown represents part of the situation. Reproduce this tree and place the probabilities on the branches.
2. Prove that $P(E) = 0.363$.
3. Given that the circuit is operational after 6000 hours, calculate the probability that valve $V_1$ is operational at that time. Round to the nearest thousandth.

Exercise 3 — Candidates who have followed the specialization
Each young parent uses only one brand of baby food jars each month. Three brands $\mathrm { X } , \mathrm { Y }$ and Z share the market. Let $n$ be a natural integer. We denote: $\quad X _ { n }$ the event ``brand X is used in month $n$ '', $Y _ { n }$ the event ``brand Y is used in month $n$ '', $Z _ { n }$ the event ``brand Z is used in month $n$ ''. The probabilities of events $X _ { n } , Y _ { n } , Z _ { n }$ are denoted respectively $x _ { n } , y _ { n } , z _ { n }$. The advertising campaign of each brand causes the distribution to change.
A buyer of brand X in month $n$ has the following month: $50 \%$ chance of remaining loyal to this brand, $40 \%$ chance of buying brand Y, $10 \%$ chance of buying brand $Z$.
A buyer of brand Y in month $n$ has the following month: $30 \%$ chance of remaining loyal to this brand, $50 \%$ chance of buying brand X, $20 \%$ chance of buying brand $Z$.
A buyer of brand Z in month $n$ has the following month: $70 \%$ chance of remaining loyal to this brand, $10 \%$ chance of buying brand X, $20 \%$ chance of buying brand Y.

1. a. Express $x _ { n + 1 }$ as a function of $x _ { n } , y _ { n }$ and $z _ { n }$.

We admit that: $y _ { n + 1 } = 0.4 x _ { n } + 0.3 y _ { n } + 0.2 z _ { n }$ and that $z _ { n + 1 } = 0.1 x _ { n } + 0.2 y _ { n } + 0.7 z _ { n }$. b. Express $z _ { n }$ as a function of $x _ { n }$ and $y _ { n }$. Deduce the expression of $x _ { n + 1 }$ and $y _ { n + 1 }$ as functions of $x _ { n }$ and $y _ { n }$.
2. We define the sequence $\left( U _ { n } \right)$ by $U _ { n } = \binom { x _ { n } } { y _ { n } }$ for every natural integer $n$.
We admit that, for every natural integer $n$, $U _ { n + 1 } = A \times U _ { n } + B$ where $A = \left( \begin{array} { l l } 0.4 & 0.4 \\ 0.2 & 0.1 \end{array} \right)$ and $B = \binom { 0.1 } { 0.2 }$.
At the beginning of the statistical study (January 2014: $n = 0$), we estimate that $U _ { 0 } = \binom { 0.5 } { 0.3 }$. Consider the following algorithm:

Variables	\begin{tabular}{l} $n$ and $i$ natural integers.
$A$, $B$ and $U$ matrices

\hline Input and initialization &

Request the value of $n$ $i$ takes the value 0

$A$ takes the value $\left( \begin{array} { l l } 0.4 & 0.4 \\ 0.2 & 0.1 \end{array} \right)$

$B$ takes the value $\binom { 0.1 } { 0.2 }$

$U$ takes the value $\binom { 0.5 } { 0.3 }$

\hline Processing &

While $i < n$

$U$ takes the value $A \times U + B$

$i$ takes the value $i + 1$

End while

\hline Output & Display $U$ \hline \end{tabular}
a. Give the results displayed by this algorithm for $n = 1$ then for $n = 3$. b. What is the probability of using brand X in April?
In the rest of the exercise, we seek to determine an expression of $U _ { n }$ as a function of $n$. We denote by $I$ the matrix $\left( \begin{array} { l l } 1 & 0 \\ 0 & 1 \end{array} \right)$ and $N$ the matrix $I - A$.
3. We denote by $C$ a column matrix with two rows. a. Prove that $C = A \times C + B$ is equivalent to $N \times C = B$. b. We admit that $N$ is an invertible matrix and that $N ^ { - 1 } = \left( \begin{array} { l l } \frac { 45 } { 23 } & \frac { 20 } { 23 } \\ \frac { 10 } { 23 } & \frac { 30 } { 23 } \end{array} \right)$.
Deduce that $C = \binom { \frac { 17 } { 46 } } { \frac { 7 } { 23 } }$.
4. We denote by $V _ { n }$ the matrix such that $V _ { n } = U _ { n } - C$ for every natural integer $n$. a. Show that, for every natural integer $n$, $V _ { n + 1 } = A \times V _ { n }$. b. We admit that $U _ { n } = A ^ { n } \times \left( U _ { 0 } - C \right) + C$.
What are the probabilities of using brands $\mathrm { X } , \mathrm { Y }$ and Z in May?

Exercise 1 — 5 points Theme: probability, sequences
Parts A and B can be treated independently
Part A
Each day, an athlete must jump over a hurdle at the end of training. Based on the previous season, his coach estimates that

• if the athlete clears the hurdle one day, then he will clear it in $90\%$ of cases the next day;
• if the athlete does not clear the hurdle one day, then in $70\%$ of cases he will not clear it the next day either.

For every natural integer $n$, we denote:

• $R_{n}$ the event: ``The athlete successfully clears the hurdle during the $n$-th session'',
• $p_{n}$ the probability of event $R_{n}$. We consider that $p_{0} = 0.6$.

1. Let $n$ be a natural integer, copy the weighted tree below and complete the blanks.
2. Justify using the tree that, for every natural integer $n$, we have: $$p_{n+1} = 0.6 p_{n} + 0.3 .$$
3. Consider the sequence $(u_{n})$ defined, for every natural integer $n$, by $u_{n} = p_{n} - 0.75$. a. Prove that the sequence $(u_{n})$ is a geometric sequence and specify its common ratio and first term. b. Prove that, for every natural integer $n$: $$p_{n} = 0.75 - 0.15 \times 0.6^{n} .$$ c. Deduce that the sequence $(p_{n})$ is convergent and determine its limit $\ell$. d. Interpret the value of $\ell$ in the context of the exercise.

Part B
After many training sessions, the coach now estimates that the athlete clears each hurdle with a probability of 0.75 and this independently of whether or not he cleared the previous hurdles. We denote $X$ the random variable that gives the number of hurdles cleared by the athlete at the end of a 400 metres hurdles race which has 10 hurdles.

1. Specify the nature and parameters of the probability distribution followed by $X$.
2. Determine, to $10^{-3}$ near, the probability that the athlete clears all 10 hurdles.
3. Calculate $p(X \geqslant 9)$, to $10^{-3}$ near.

Exercise 2

5 points With a concern for environmental preservation, Mr. Durand decides to go to work each morning using his bicycle or public transport. If he chooses to take public transport one morning, he takes public transport again the next day with a probability equal to 0.8. If he uses his bicycle one morning, he uses his bicycle again the next day with a probability equal to 0.4. For every non-zero natural number $n$, we denote:

• $T _ { n }$ the event ``Mr. Durand uses public transport on the $n$-th day''
• $V _ { n }$ the event ``Mr. Durand uses his bicycle on the $n$-th day''
• We denote $p _ { n }$ the probability of the event $T _ { n }$,

On the first morning, he decides to use public transport. Thus, the probability of the event $T _ { 1 }$ is $p _ { 1 } = 1$.

1. Copy and complete the probability tree below representing the situation for the $2 ^ { \mathrm { nd } }$ and $3 ^ { \mathrm { rd } }$ days.
2. Calculate $p _ { 3 }$
3. On the $3 ^ { \mathrm { rd } }$ day, Mr. Durand uses his bicycle. Calculate the probability that he took public transport the day before.
4. Copy and complete the probability tree below representing the situation for the $n$-th and ( $n + 1$ )-th days.
5. Show that, for every non-zero natural number $n$, $p _ { n + 1 } = 0,2 p _ { n } + 0,6$.
6. Show by induction that, for every non-zero natural number $n$, we have $$p _ { n } = 0,75 + 0,25 \times 0,2 ^ { n - 1 } .$$
7. Determine the limit of the sequence ( $p _ { n }$ ) and interpret the result in the context of the exercise.

A car dealership sells two types of vehicles:

• $60\%$ are fully electric vehicles;
• $40\%$ are rechargeable hybrid vehicles.

$75\%$ of buyers of fully electric vehicles and $52\%$ of buyers of rechargeable hybrid vehicles have the material possibility of installing a charging station at home.
A buyer is chosen at random and the following events are considered:

• $E$: ``the buyer chooses a fully electric vehicle'';
• $B$: ``the buyer has the possibility of installing a charging station at home''.

Throughout the exercise, probabilities should be rounded to the nearest thousandth if necessary.

1. Calculate the probability that the buyer chooses a fully electric vehicle and has the possibility of installing a charging station at home.
   A weighted tree diagram may be used.
2. Prove that $P(B) = 0.658$.
3. A buyer has the possibility of installing a charging station at home. What is the probability that he chooses a fully electric vehicle?
4. A sample of 20 buyers is chosen. This sampling is treated as drawing with replacement. Let $X$ be the random variable that gives the total number of buyers able to install a charging station at home among the sample of 20 buyers. a. Determine the nature and parameters of the probability distribution followed by $X$. b. Calculate $P(X = 8)$. c. Calculate the probability that at least 10 buyers can install a charging station. d. Calculate the expected value of $X$. e. The dealership manager decides to offer the installation of the charging station to buyers who have the possibility of installing one at home. This installation costs $1200$~\euro. On average, what amount should she plan to spend on this offer when selling 20 vehicles?

Léa spends a good part of her days playing a video game and is interested in the chances of winning her next games.
She estimates that if she has just won a game, she wins the next one in $70\%$ of cases. But if she has just suffered a defeat, according to her, the probability that she wins the next one is 0.2. Furthermore, she thinks she has an equal chance of winning the first game as of losing it.
For all non-zero natural integer $n$, we define the following events:

• $G _ { n }$: ``Léa wins the $n$-th game of the day'';
• $D _ { n }$: ``Léa loses the $n$-th game of the day''.

For all non-zero natural integer $n$, we denote $g _ { n }$ the probability of event $G _ { n }$. We have therefore $g _ { 1 } = 0.5$.

1. What is the value of the conditional probability $p _ { G _ { 1 } } \left( D _ { 2 } \right)$?
2. Copy and complete the probability tree below which models the situation for the first two games of the day.
3. Calculate $g _ { 2 }$.
4. Let $n$ be a non-zero natural integer. a. Copy and complete the probability tree below which models the situation for the $n$-th and $(n+1)$-th games of the day. b. Justify that for all non-zero natural integer $n$, $$g _ { n + 1 } = 0.5 g _ { n } + 0.2 .$$
5. For all non-zero natural integer $n$, we set $v _ { n } = g _ { n } - 0.4$. a. Show that the sequence $( v _ { n } )$ is geometric. We will specify its first term and its common ratio. b. Show that, for all non-zero natural integer $n$: $$g _ { n } = 0.1 \times 0.5 ^ { n - 1 } + 0.4 .$$
6. Study the variations of the sequence $( g _ { n } )$.
7. Give, by justifying, the limit of the sequence $( g _ { n } )$. Interpret the result in the context of the problem.
8. Determine, by calculation, the smallest integer $n$ such that $g _ { n } - 0.4 \leqslant 0.001$.
9. Copy and complete lines 4, 5 and 6 of the following function, written in Python language, so that it returns the smallest rank from which the terms of the sequence $\left( g _ { n } \right)$ are all less than or equal to $0.4 + e$, where $e$ is a strictly positive real number. \begin{verbatim} def seuil(e) : g = 0.5 n = 1 while...: g = 0.5 * g + 0.2 n = ... return (n) \end{verbatim}

A robot is positioned on a horizontal axis and moves several times by one meter on this axis, randomly to the right or to the left. During the first movement, the probability that the robot moves to the right is equal to $\frac{1}{3}$. If it moves to the right, the probability that the robot moves to the right again during the next movement is equal to $\frac{3}{4}$. If it moves to the left, the probability that the robot moves to the left again during the next movement is equal to $\frac{1}{2}$. For every natural integer $n \geqslant 1$, we denote:

• $D_n$ the event: ``the robot moves to the right during the $n$-th movement'';
• $\overline{D_n}$ the complementary event of $D_n$;
• $p_n$ the probability of event $D_n$.

We therefore have $p_1 = \frac{1}{3}$.
Part A: study of the special case where $n = 2$ In this part, the robot performs two successive movements.

1. Reproduce and complete the following weighted tree.
2. Determine the probability that the robot moves to the right twice.
3. Show that $p_2 = \frac{7}{12}$.
4. The robot moved to the left during the second movement. What is the probability that it moved to the right during the first movement?

Part B: study of the sequence $(p_n)$. We wish to estimate the movement of the robot after a large number of steps.

1. Prove that for every natural integer $n \geqslant 1$, we have: $$p_{n+1} = \frac{1}{4} p_n + \frac{1}{2}.$$ You may use a tree to help.
2. a. Show by induction that for every natural integer $n \geqslant 1$, we have: $$p_n \leqslant p_{n+1} < \frac{2}{3}.$$ b. Is the sequence $(p_n)$ convergent? Justify.
3. We consider the sequence $(u_n)$ defined for every natural integer $n \geqslant 1$, by $u_n = p_n - \frac{2}{3}$. a. Show that the sequence $(u_n)$ is geometric and specify its first term and its common ratio. b. Determine the limit of the sequence $(p_n)$ and interpret the result in the context of the exercise.

Part C In this part, we consider another robot that performs ten movements of one meter independent of each other, each movement to the right having a fixed probability equal to $\frac{3}{4}$. What is the probability that it returns to its starting point after the ten movements? Round the result to $10^{-3}$ near.

Em um colégio, a probabilidade de um aluno ser aprovado em Matemática é 0,7 e a probabilidade de ser aprovado em Português é 0,8. Assumindo que as aprovações são eventos independentes, a probabilidade de um aluno ser aprovado em ambas as disciplinas é
(A) 0,14 (B) 0,24 (C) 0,56 (D) 0,75 (E) 0,90

Any two events $X$ and $Y$ are called mutually exclusive when the probability $P(X$ and $Y) = 0$ and they are called exhaustive when $P(X$ or $Y) = 1$. Suppose $A$ and $B$ are events and the probability of each of these two events is strictly between 0 and 1 (i.e., $0 < P(A) < 1$ and $0 < P(B) < 1$).
Statements
(5) $A$ and $B$ are mutually exclusive if and only if not $A$ and not $B$ are exhaustive. (6) $A$ and $B$ are independent if and only if not $A$ and not $B$ are independent. (7) $A$ and $B$ cannot be simultaneously independent and exhaustive. (8) $A$ and $B$ cannot be simultaneously mutually exclusive and exhaustive.

For two independent events $A$ and $B$, $$\mathrm { P } ( A \cap B ) = 2 \mathrm { P } \left( A \cap B ^ { c } \right) , \quad \mathrm { P } \left( A ^ { c } \cap B \right) = \frac { 1 } { 12 }$$ find the value of $\mathrm { P } ( A )$. (Given that $\mathrm { P } ( A ) \neq 0$.) [3 points]
(1) $\frac { 1 } { 2 }$
(2) $\frac { 5 } { 8 }$
(3) $\frac { 3 } { 4 }$
(4) $\frac { 7 } { 8 }$
(5) $\frac { 15 } { 16 }$

Two events $A , B$ are independent and $\mathrm { P } \left( A ^ { C } \right) = \mathrm { P } ( B ) = \frac { 1 } { 3 }$. What is the value of $\mathrm { P } ( A \cap B )$? (Here, $A ^ { C }$ is the complement of $A$.) [3 points]
(1) $\frac { 1 } { 18 }$
(2) $\frac { 1 } { 9 }$
(3) $\frac { 1 } { 6 }$
(4) $\frac { 2 } { 9 }$
(5) $\frac { 5 } { 18 }$

Two events $A$ and $B$ are mutually exclusive, and $$\mathrm { P } ( A ) = \mathrm { P } ( B ) , \quad \mathrm { P } ( A ) \mathrm { P } ( B ) = \frac { 1 } { 9 }$$ What is the value of $\mathrm { P } ( A \cup B )$? [3 points]
(1) $\frac { 1 } { 6 }$
(2) $\frac { 1 } { 3 }$
(3) $\frac { 1 } { 2 }$
(4) $\frac { 2 } { 3 }$
(5) $\frac { 5 } { 6 }$

Two events $A$ and $B$ are mutually independent, and $$\mathrm { P } ( A ) = \frac { 2 } { 3 } , \mathrm { P } ( A \cap B ) = \mathrm { P } ( A ) - \mathrm { P } ( B )$$ What is the value of $\mathrm { P } ( B )$? [3 points]
(1) $\frac { 1 } { 10 }$
(2) $\frac { 1 } { 5 }$
(3) $\frac { 3 } { 10 }$
(4) $\frac { 2 } { 5 }$
(5) $\frac { 1 } { 2 }$

A student named Chulsu participated in a design competition. Participants receive scores in two categories, and the possible scores in each category are one of three types shown in the table. The probability that Chulsu receives score A in each category is $\frac { 1 } { 2 }$, the probability of receiving score B is $\frac { 1 } { 3 }$, and the probability of receiving score C is $\frac { 1 } { 6 }$. When the event of receiving audience voting scores and the event of receiving judge scores are mutually independent, what is the probability that the sum of the two scores Chulsu receives is 70? [3 points]

Category	Score A	Score B	Score C
Audience Voting	40	30	20
Judges	50	40	30

(1) $\frac { 1 } { 3 }$
(2) $\frac { 11 } { 36 }$
(3) $\frac { 5 } { 18 }$
(4) $\frac { 1 } { 4 }$
(5) $\frac { 2 } { 9 }$

Two events $A$ and $B$ are independent, and
$$\mathrm { P } ( A \cup B ) = \frac { 1 } { 2 } , \quad \mathrm { P } ( A \mid B ) = \frac { 3 } { 8 }$$
What is the value of $\mathrm { P } \left( A \cap B ^ { C } \right)$? (where $B ^ { C }$ is the complement of $B$) [3 points]
(1) $\frac { 1 } { 10 }$
(2) $\frac { 3 } { 20 }$
(3) $\frac { 1 } { 5 }$
(4) $\frac { 1 } { 4 }$
(5) $\frac { 3 } { 10 }$

Two events $A , B$ are independent, and $\mathrm { P } ( A ) = \frac { 1 } { 3 } , \mathrm { P } ( B ) = \frac { 1 } { 3 }$. What is the value of $\mathrm { P } \left( A \cap B ^ { C } \right)$? (Here, $B ^ { C }$ is the complement of $B$.) [3 points]
(1) $\frac { 5 } { 27 }$
(2) $\frac { 2 } { 9 }$
(3) $\frac { 7 } { 27 }$
(4) $\frac { 8 } { 27 }$
(5) $\frac { 1 } { 3 }$

Two events $A$ and $B$ are independent, and $$\mathrm { P } \left( A ^ { C } \right) = \frac { 1 } { 4 } , \quad \mathrm { P } ( A \cap B ) = \frac { 1 } { 2 }$$ What is the value of $\mathrm { P } \left( B \mid A ^ { C } \right)$? (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) $\frac { 3 } { 4 }$

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

Two events $A$ and $B$ are mutually independent and $$\mathrm { P } ( A ) = \frac { 2 } { 3 } , \quad \mathrm { P } ( A \cup B ) = \frac { 5 } { 6 }$$ What is the value of $\mathrm { P } ( B )$? [3 points]
(1) $\frac { 1 } { 3 }$
(2) $\frac { 5 } { 12 }$
(3) $\frac { 1 } { 2 }$
(4) $\frac { 7 } { 12 }$
(5) $\frac { 2 } { 3 }$

Two events $A$ and $B$ are independent, and $$\mathrm { P } ( A ) = \frac { 2 } { 3 } , \quad \mathrm { P } ( A \cup B ) = \frac { 5 } { 6 }$$ Find the value of $\mathrm { P } ( B )$. [3 points]
(1) $\frac { 1 } { 3 }$
(2) $\frac { 5 } { 12 }$
(3) $\frac { 1 } { 2 }$
(4) $\frac { 7 } { 12 }$
(5) $\frac { 2 } { 3 }$

A die is rolled once. Let A be the event that an odd number appears, and for a natural number $m$ with $m \leq 6$, let B be the event that a divisor of $m$ appears. Find the sum of all values of $m$ such that the two events A and B are independent. [4 points]

Two events $A$ and $B$ are independent and $$\mathrm { P } ( A \mid B ) = \mathrm { P } ( B ) , \quad \mathrm { P } ( A \cap B ) = \frac { 1 } { 9 }$$ What is the value of $\mathrm { P } ( A )$? [3 points]
(1) $\frac { 7 } { 18 }$
(2) $\frac { 1 } { 3 }$
(3) $\frac { 5 } { 18 }$
(4) $\frac { 2 } { 9 }$
(5) $\frac { 1 } { 6 }$

Two events $A$ and $B$ are independent, and $$\mathrm{P}(A \cap B) = \frac{1}{4}, \quad \mathrm{P}(A^C) = 2\mathrm{P}(A)$$ Find the value of $\mathrm{P}(B)$. (Here, $A^C$ is the complement of $A$.) [3 points]
(1) $\frac{3}{8}$
(2) $\frac{1}{2}$
(3) $\frac{5}{8}$
(4) $\frac{3}{4}$
(5) $\frac{7}{8}$

Let $x$ be a real number such that $x > 1$ and let $X$ be a random variable that follows the zeta distribution with parameter $x$, i.e. $$\forall n \in \mathbb{N}^{*}, \quad \mathbb{P}(X = n) = \frac{1}{\zeta(x) n^{x}}$$ Let $(q_{1}, \ldots, q_{n}) \in \mathcal{P}^{n}$ be an $n$-tuple of distinct prime numbers. Show that the events $(X \in q_{1}\mathbb{N}^{*}), \ldots, (X \in q_{n}\mathbb{N}^{*})$ are mutually independent.

Let $x \in \mathbb{R}$ such that $x > 1$. Let $X$ and $Y$ be two independent random variables each following a zeta probability distribution with parameter $x$, i.e. $$\forall n \in \mathbb{N}^{*}, \quad \mathbb{P}(X = n) = \mathbb{P}(Y = n) = \frac{1}{\zeta(x) n^{x}}$$ Let $A$ be the event ``No prime number divides $X$ and $Y$ simultaneously''. For all $n \in \mathbb{N}^{*}$, denote by $C_{n}$ the event $$C_{n} = \bigcap_{k=1}^{n} \left((X \notin p_{k}\mathbb{N}^{*}) \cup (Y \notin p_{k}\mathbb{N}^{*})\right)$$ Express the event $A$ using the events $C_{n}$. Deduce that $$\mathbb{P}(A) = \frac{1}{\zeta(2x)}$$