Exercise 4 (5 points) — Candidates who have not followed the specialization course
Each week, a farmer offers for direct sale to each of his customers a basket of fresh products that contains a single bottle of fruit juice. A statistical study carried out gives the following results:

• at the end of the first week, the probability that a customer returns the bottle from his basket is 0.9;
• if the customer returned the bottle from his basket one week, then the probability that he brings back the bottle from the basket the following week is 0.95;
• if the customer did not return the bottle from his basket one week, then the probability that he brings back the bottle from the basket the following week is 0.2.

A customer is chosen at random from the farmer's clientele. For any non-zero natural number $n$, we denote by $R_n$ the event ``the customer returns the bottle from his basket in the $n$-th week''.
1. a. Model the situation studied for the first two weeks using a weighted tree that will involve the events $R_1$ and $R_2$.
b. Determine the probability that the customer returns the bottles from the baskets of the first and second weeks.
c. Show that the probability that the customer returns the bottle from the basket of the second week is equal to 0.875.
d. Given that the customer returned the bottle from his basket in the second week, what is the probability that he did not return the bottle from his basket in the first week? Round the result to $10^{-3}$.
2. For any non-zero natural number $n$, we denote by $r_n$ the probability that the customer returns the bottle from the basket in the $n$-th week. We then have $r_n = p(R_n)$.
a. Copy and complete the weighted tree (no justification is required).
b. Justify that for any non-zero natural number $n$, $r_{n+1} = 0.75r_n + 0.2$.
c. Prove that for any non-zero natural number $n$, $r_n = 0.1 \times 0.75^{n-1} + 0.8$.
d. Calculate the limit of the sequence $(r_n)$. Interpret the result in the context of the exercise.

Exercise 1 (7 points) Theme: probabilities, sequences In a tourist region, a company offers a bicycle rental service for the day. The company has two distinct rental points, point A and point B. Bicycles can be borrowed and returned indifferently at either of the two rental points. It is assumed that the total number of bicycles is constant and that every morning, when the service opens, each bicycle is at point A or point B. According to a statistical study:

• If a bicycle is at point A one morning, the probability that it is at point A the next morning is equal to 0.84;
• If a bicycle is at point B one morning, the probability that it is at point B the next morning is equal to 0.76.

When the service opens on the first morning, the company has placed half of its bicycles at point A and the other half at point B. We consider a bicycle from the company chosen at random. For every positive integer $n$, the following events are defined:

• $A _ { n }$ : ``the bicycle is at point A on the $n$-th morning''
• $B _ { n }$ : ``the bicycle is at point B on the $n$-th morning''.

For every positive integer $n$, we denote by $a _ { n }$ the probability of event $A _ { n }$ and by $b _ { n }$ the probability of event $B _ { n }$. Thus $a _ { 1 } = 0.5$ and $b _ { 1 } = 0.5$.

1. Copy and complete the weighted tree that models the situation for the first two mornings.
2. a. Calculate $a _ { 2 }$. b. The bicycle is at point A on the second morning. Calculate the probability that it was at point B on the first morning. The probability will be rounded to the nearest thousandth.
3. a. Copy and complete the weighted tree that models the situation for the $n$-th and $(n + 1)$-th mornings. b. Justify that for every positive integer $n$, $a _ { n + 1 } = 0.6 a _ { n } + 0.24$.
4. Show by induction that, for every positive integer $n$, $a _ { n } = 0.6 - 0.1 \times 0.6 ^ { n - 1 }$.
5. Determine the limit of the sequence $(a _ { n })$ and interpret this limit in the context of the exercise.
6. Determine the smallest positive integer $n$ such that $a _ { n } \geqslant 0.599$ and interpret the result obtained in the context of the exercise.

A game offered at a fairground consists of making three successive shots at a moving target.
It has been observed that:

• If the player hits the target on one shot then they miss it on the next shot in $65\%$ of cases;
• If the player misses the target on one shot then they hit it on the next shot in $50\%$ of cases.

The probability that a player hits the target on their first shot is 0.6. For any event $A$, we denote $p(A)$ its probability and $\bar{A}$ the complementary event of $A$. We randomly choose a player for this shooting game. We consider the following events:

• $A_1$: ``The player hits the target on the $1^{\text{st}}$ shot''
• $A_2$: ``The player hits the target on the $2^{\mathrm{nd}}$ shot''
• $A_3$: ``The player hits the target on the $3^{\mathrm{rd}}$ shot''.

Part A

1. Copy and complete, with the corresponding probabilities on each branch, the probability tree below modelling the situation.

Let $X$ be the random variable that gives the number of times the player hits the target during the three shots.
2. Show that the probability that the player hits the target exactly twice during the three shots is equal to 0.4015.
3. The objective of this question is to calculate the expectation of the random variable $X$, denoted $E(X)$. a. Copy and complete the table below giving the probability distribution of the random variable $X$.

$X = x_i$	0	1	2	3
$p\left(X = x_i\right)$	0.1			0.0735

b. Calculate $E(X)$. c. Interpret the previous result in the context of the exercise.
Part B
We consider $N$, a natural number greater than or equal to 1.
A group of $N$ people comes to this stand to play this game under identical and independent conditions.
A player is declared a winner when they hit the target three times. We denote $Y$ the random variable that counts among the $N$ people the number of players declared winners.

1. In this question, $N = 15$. a. Justify that $Y$ follows a binomial distribution and determine its parameters. b. Give the probability, rounded to $10^{-3}$, that exactly 5 players win this game.
2. By the method of your choice, which you will explain, determine the minimum number of people who must come to this stand so that the probability that there is at least one winning player is greater than or equal to 0.98.

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.

During a training session, a volleyball player practises serving. The probability that he succeeds on the first serve is equal to 0.85.
We further assume that the following two conditions are satisfied:

• if the player succeeds on a serve, then the probability that he succeeds on the next one is equal to 0.6;
• if the player fails a serve, then the probability that he fails the next one is equal to 0.6.

For any non-zero natural number $n$, we denote by $R _ { n }$ the event ``the player succeeds on the $n$-th serve'' and $\overline { R _ { n } }$ the complementary event.
Part A We are interested in the first two serves of the training session.

1. Represent the situation with a probability tree.
2. Prove that the probability of event $R _ { 2 }$ is equal to 0.57.
3. Given that the player succeeded on the second serve, calculate the probability that he failed the first one.
4. Let $Z$ be the random variable equal to the number of successful serves during the first two serves. a. Determine the probability distribution of $Z$ (you may use the probability tree from question 1). b. Calculate the mathematical expectation $\mathrm { E } ( Z )$ of the random variable $Z$.

Interpret this result in the context of the exercise.
Part B We now consider the general case. For any non-zero natural number $n$, we denote by $x _ { n }$ the probability of event $R _ { n }$.

1. a. Give the conditional probabilities $P _ { R _ { n } } \left( R _ { n + 1 } \right)$ and $P _ { \overline { R _ { n } } } \left( \overline { R _ { n + 1 } } \right)$. b. Show that, for any non-zero natural number $n$, we have: $x _ { n + 1 } = 0.2 x _ { n } + 0.4$.
2. Let the sequence $(u _ { n })$ be defined for any non-zero natural number $n$ by: $u _ { n } = x _ { n } - 0.5$. a. Show that the sequence $(u _ { n })$ is a geometric sequence. b. Determine the expression of $x _ { n }$ as a function of $n$. Deduce the limit of the sequence $\left( x _ { n } \right)$. c. Interpret this limit in the context of the exercise.

A student eats every day at the university restaurant. This restaurant offers vegetarian and non-vegetarian dishes.

• When on a given day the student has chosen a vegetarian dish, the probability that he chooses a vegetarian dish the next day is 0.9.
• When on a given day the student has chosen a non-vegetarian dish, the probability that he chooses a vegetarian dish the next day is 0.7.

For any natural number $n$, we denote by $V _ { n }$ the event ``the student chose a vegetarian dish on the $n ^ { \mathrm { th } }$ day'' and $p _ { n }$ the probability of $V _ { n }$. On the first day of the semester, the student chose the vegetarian dish. Thus $p _ { 1 } = 1$.

1. a. Indicate the value of $p _ { 2 }$. b. Show that $p _ { 3 } = 0.88$. You may use a probability tree. c. Given that on the 3rd day the student chose a vegetarian dish, what is the probability that he chose a non-vegetarian dish the previous day? Round the result to $10 ^ { - 2 }$.
2. Copy and complete the probability tree.
3. Justify that, for any natural number $n \geqslant 1 , p _ { n + 1 } = 0.2 p _ { n } + 0.7$.
4. We wish to have the list of the first terms of the sequence $( p _ { n } )$ for $n \geqslant 1$. For this, we use a function called meals programmed in Python language, of which three versions are proposed below.

\begin{verbatim} Program 1 def meals(n): p=1 L= [p] for k in range(1,n): p = 0.2*p+0.7 L. append(p) return(L) \end{verbatim}
\begin{verbatim} Program 2 def meals(n): p=1 L= [p] for k in range(1,n+1): p = 0.2*p+0.7 L. append(p) return(L) \end{verbatim}
\begin{verbatim} Program 3 def meals(n): p=1 L=[p] for k in range(1,n): p = 0.2*p+0.7 L.append(p+1) return(L) \end{verbatim}
a. Which of these programs allows displaying the first $n$ terms of the sequence $\left( p _ { n } \right)$? No justification is required. b. With the program chosen in question a., give the result displayed for $n = 5$.
4. Prove by induction that, for any natural number $n \geqslant 1 , p _ { n } = 0.125 \times 0.2 ^ { n - 1 } + 0.875$.
5. Deduce the limit of the sequence $\left( p _ { n } \right)$.