Part B

Between two phases of the competition, to improve, the competitor works on his lateral precision on another training target. He shoots arrows to try to hit a vertical band, with width 20 cm (shaded in the figure), as close as possible to the central vertical line. The plane containing the vertical band is equipped with a coordinate system: the central line aimed at is the $y$-axis. Let $X$ denote the random variable that, for any arrow shot reaching this plane, associates the abscissa of its point of impact.
It is assumed that the random variable $X$ follows a normal distribution with mean 0 and standard deviation 10.

1. When the arrow reaches the plane, determine the probability that its point of impact is located outside the shaded band.
2. How should the edges of the shaded band be modified so that, when the arrow reaches the plane, its point of impact is located inside the band with a probability equal to 0.6?

Sofia wishes to go to the cinema. She can go by bike or by bus.

Part A: Using the bus

We assume in this part that Sofia uses the bus to go to the cinema. The duration of the journey between her home and the cinema (expressed in minutes) is modelled by the random variable $T _ { B }$ which follows the uniform distribution on [12; 15].

1. Prove that the probability that Sofia takes between 12 and 14 minutes is $\frac { 2 } { 3 }$.
2. Give the average duration of the journey.

Part B: Using her bike

We now assume that Sofia chooses to use her bike. The duration of the journey (expressed in minutes) is modelled by the random variable $T _ { v }$ which follows the normal distribution with mean $\mu = 14$ and standard deviation $\sigma = 1,5$.

1. What is the probability that Sofia takes less than 14 minutes to go to the cinema? What is the probability that Sofia takes between 12 and 14 minutes to go to the cinema? Round the result to $10 ^ { - 3 }$.

Part C: Playing with dice

Sofia is hesitating between the bus and the bike. She decides to roll a fair 6-sided die. If she gets 1 or 2, she takes the bus, otherwise she takes her bike. We denote:

• $B$ the event ``Sofia takes the bus'';
• $V$ the event ``Sofia takes her bike'';
• C the event ``Sofia takes between 12 and 14 minutes to go to the cinema''.

1. Prove that the probability, rounded to $10 ^ { - 2 }$, that Sofia takes between 12 and 14 minutes is 0.49.
2. Given that Sofia took between 12 and 14 minutes to go to the cinema, what is the probability, rounded to $10 ^ { - 2 }$, that she used the bus?

A fruit and vegetable retailer buys melons from market gardener A. The mass in grams of melons from market gardener A is modelled by a random variable $M_\mathrm{A}$ that follows a uniform distribution on the interval $[850; x]$, where $x$ is a real number greater than 1200. Melons are described as ``compliant'' if their mass is between 900 g and 1200 g. The retailer observes that $75\%$ of melons from market gardener A are compliant. Determine $x$.

Exercise 2 (5 points)

An online platform offers two types of video games: a game of type $A$ and a game of type $B$.

Part A

The durations of games of type $A$ and type $B$, expressed in minutes, can be modeled respectively by two random variables $X_A$ and $X_B$. The random variable $X_A$ follows the uniform distribution on the interval $[9; 25]$. The random variable $X_B$ follows the normal distribution with mean $\mu$ and standard deviation 3.

1. a. Calculate the average duration of a game of type $A$. b. Specify using the graph the average duration of a game of type $B$.
2. We choose at random, with equal probability, a game type. What is the probability that the duration of a game is less than 20 minutes? Give the result rounded to the nearest hundredth.

Part B

It is admitted that, as soon as the player completes a game, the platform proposes a new game according to the following model:

• if the player completes a game of type $A$, the platform proposes to play again a game of type $A$ with probability 0.8;
• if the player completes a game of type $B$, the platform proposes to play again a game of type $B$ with probability 0.7.

For a natural number $n$ greater than or equal to 1, we denote $A_n$ and $B_n$ the events: $A_n$: ``the $n$-th game is a game of type $A$.'' $B_n$: ``the $n$-th game is a game of type $B$.'' For any natural number $n$ greater than or equal to 1, we denote $a_n$ the probability of event $A_n$.

1. a. Copy and complete the probability tree. b. Show that for any natural number $n \geqslant 1$, we have: $a_{n+1} = 0.5\,a_n + 0.3$.

In the rest of the exercise, we denote $a$ the probability that the player plays game $A$ during his first game, where $a$ is a real number belonging to the interval $[0; 1]$. The sequence $(a_n)$ is therefore defined by: $a_1 = a$, and for any natural number $n \geqslant 1$, $a_{n+1} = 0.5\,a_n + 0.3$.

1. Study of a particular case. In this question, we assume that $a = 0.5$. a. Show by induction that for any natural number $n \geqslant 1$, we have: $0 \leqslant a_n \leqslant 0.6$. b. Show that the sequence $(a_n)$ is increasing. c. Show that the sequence $(a_n)$ is convergent and specify its limit.
2. Study of the general case. In this question, the real number $a$ belongs to the interval $[0; 1]$. We consider the sequence $(u_n)$ defined for any natural number $n \geqslant 1$ by $u_n = a_n - 0.6$. a. Show that the sequence $(u_n)$ is a geometric sequence. b. Deduce that for any natural number $n \geqslant 1$, we have: $a_n = (a - 0.6) \times 0.5^{n-1} + 0.6$. c. Determine the limit of the sequence $(a_n)$. Does this limit depend on the value of $a$? d. The platform broadcasts an advertisement inserted at the beginning of games of type $A$ and another inserted at the beginning of games of type $B$. Which advertisement should be the most viewed by a player intensively playing video games?

Let $X$ denote a random variable following the uniform distribution on $\left[ 0 ; \frac { \pi } { 2 } \right]$. The probability that a value taken by the random variable $X$ is a solution to the inequality $\cos x > \frac { 1 } { 2 }$ is equal to:
Answer A: $\frac { 2 } { 3 } \quad$ Answer B: $\frac { 1 } { 3 } \quad$ Answer C: $\frac { 1 } { 2 } \quad$ Answer D: $\frac { 1 } { \pi }$

Let $(U_n)_{n \geqslant 1}$ be a sequence of mutually independent random variables following a Bernoulli distribution with parameter $1/2$. We set $Y_n = \sum_{k=1}^{n} \frac{U_k}{2^k}$.
Establish, for every non-zero natural number $n$, that $Y_n$ follows a uniform distribution on $D_n$.

Let $n$ be a non-zero natural number and let $X_n$ be a random variable that follows a uniform distribution on $D_n$. Show that there exist random variables $V_1, \ldots, V_n$ mutually independent, each following a Bernoulli distribution with parameter $1/2$, and such that $$X_n = \sum_{k=1}^{n} \frac{V_k}{2^k}.$$

Let $(U_n)_{n \geqslant 1}$ be a sequence of mutually independent random variables following a Bernoulli distribution with parameter $1/2$. We set $Y_n = \sum_{k=1}^{n} \frac{U_k}{2^k}$, $F_n(x) = \mathbb{P}(Y_n \leqslant x)$ and $G_n(x) = \mathbb{P}(Y_n < x)$.
Let $x$ be a real number. Establish the monotonicity of the sequences $(F_n(x))_{n \geqslant 1}$ and $(G_n(x))_{n \geqslant 1}$.

Let $(U_n)_{n \geqslant 1}$ be a sequence of mutually independent random variables following a Bernoulli distribution with parameter $1/2$. We set $Y_n = \sum_{k=1}^{n} \frac{U_k}{2^k}$, $F_n(x) = \mathbb{P}(Y_n \leqslant x)$ and $G_n(x) = \mathbb{P}(Y_n < x)$.
Using the monotonicity established in Q24, deduce the pointwise convergence of the sequences of functions $(F_n)_{n \geqslant 1}$ and $(G_n)_{n \geqslant 1}$.

Let $(U_n)_{n \geqslant 1}$ be a sequence of mutually independent random variables following a Bernoulli distribution with parameter $1/2$. We set $Y_n = \sum_{k=1}^{n} \frac{U_k}{2^k}$, $F_n(x) = \mathbb{P}(Y_n \leqslant x)$ and $G_n(x) = \mathbb{P}(Y_n < x)$, and $D = \bigcup_{n \in \mathbb{N}^{\star}} D_n$.
Show $$\forall x \in D \cup \{1\}, \quad \lim_{n \rightarrow \infty} F_n(x) = x \quad \text{and} \quad \lim_{n \rightarrow \infty} G_n(x) = x.$$

Let $(U_n)_{n \geqslant 1}$ be a sequence of mutually independent random variables following a Bernoulli distribution with parameter $1/2$. We set $Y_n = \sum_{k=1}^{n} \frac{U_k}{2^k}$, $F_n(x) = \mathbb{P}(Y_n \leqslant x)$ and $G_n(x) = \mathbb{P}(Y_n < x)$.
Generalize the results obtained in Q26 for all $x \in [0,1]$.

Let $(U_n)_{n \geqslant 1}$ be a sequence of mutually independent random variables following a Bernoulli distribution with parameter $1/2$. We set $Y_n = \sum_{k=1}^{n} \frac{U_k}{2^k}$.
Show that for every non-empty interval $I \subset [0,1]$, we have $$\lim_{n \rightarrow \infty} \mathbb{P}(Y_n \in I) = \ell(I) \quad \text{with} \quad \ell(I) = \sup I - \inf I.$$

Let $(U_n)_{n \geqslant 1}$ be a sequence of mutually independent random variables following a Bernoulli distribution with parameter $1/2$. We set $Y_n = \sum_{k=1}^{n} \frac{U_k}{2^k}$.
Using the result of Q28, deduce that for every continuous function $f$ from $[0,1]$ to $\mathbb{R}$, the sequence $(\mathbb{E}(f(Y_n)))_{n \geqslant 1}$ converges and specify its limit.

Using the result of Q29, propose another proof of the result obtained in question 6, i.e., the pointwise limit of the sequence of functions $(\varphi_n)_{n \geqslant 1}$ defined by $$\forall n \in \mathbb{N}^{\star}, \quad \varphi_n(t) = \mathbb{E}(\cos(t X_n)).$$

Let $(U_n)_{n \geqslant 1}$ be a sequence of mutually independent random variables following a Bernoulli distribution with parameter $1/2$. We set $Y_n = \sum_{k=1}^{n} \frac{U_k}{2^k}$.
Justify the existence of $\int_0^1 \frac{t-1}{\ln t} \,\mathrm{d}t$ and then determine its value.
One may consider $\int_0^1 \mathbb{E}(t^{Y_n}) \,\mathrm{d}t$.