Exercise 4 — Candidates who have followed the speciality course
We have two urns $U$ and $V$ each containing two balls. Initially, urn $U$ contains two white balls and urn $V$ contains two black balls. We perform successive draws from these urns as follows: each draw consists of taking at random, simultaneously, one ball from each urn and putting it in the other urn. For any non-zero natural number $n$, we denote by $X_n$ the random variable equal to the number of white balls in urn $U$ after $n$ draws.

Exercise 4 — Candidates who have followed the specialization course
For each of the following statements, say whether it is true or false by justifying the answer. One point is awarded for each correct justified answer. An unjustified answer will not be taken into account and the absence of an answer is not penalized.

• Consider the system $\left\{ \begin{array} { l l l l } n & \equiv & 1 & { [ 5 ] } \\ n & \equiv & 3 & { [ 4 ] } \end{array} \right.$ with unknown $n$ a relative integer.

Statement 1: If $n$ is a solution of this system then $n - 11$ is divisible by 4 and by 5. Statement 2: For all relative integer $k$, the integer $11 + 20 k$ is a solution of the system. Statement 3: If a relative integer $n$ is a solution of the system then there exists a relative integer $k$ such that $n = 11 + 20 k$.

• An automaton can be in one of two states A or B. At each second it can either remain in the state it is in or change it, with probabilities given by the probabilistic graph below. For all natural number $n$, we denote $a _ { n }$ the probability that the automaton is in state A after $n$ seconds and $b _ { n }$ the probability that the automaton is in state B after $n$ seconds. Initially, the automaton is in state B.

Consider the following algorithm:

\begin{tabular}{l} Variables: Initialization:

Processing:

Output:

&

$a$ and $b$ are real numbers

$a$ takes the value 0

$b$ takes the value 1

For $k$ going from 1 to 10

$a$ takes the value $0.8 a + 0.3 b$

$b$ takes the value $1 - a$

End For

Display $a$

Display $b$

\hline \end{tabular}
Statement 4: On output, this algorithm displays the values of $a _ { 10 }$ and $b _ { 10 }$. Statement 5: After 4 seconds, the automaton has an equal chance of being in state $A$ or being in state $B$.

Exercise 4 -- 5 points -- For candidates who have not followed the specialization course

We study a model of virus propagation in a population, week after week. Each individual in the population can be:

• either susceptible to being affected by the virus (``of type S'');
• either sick (affected by the virus);
• either immunized (cannot be affected by the virus).

For any natural integer $n$, the model of virus propagation is defined by the following rules:

• Among individuals of type S in week $n$, in week $n+1$: $85\%$ remain of type S, $5\%$ become sick and $10\%$ become immunized;
• Among sick individuals in week $n$, in week $n+1$: $65\%$ remain sick, and $35\%$ are cured and become immunized.
• Any individual immunized in week $n$ remains immunized in week $n+1$.

We randomly choose an individual from the population. We consider the following events: $S_{n}$: ``the individual is of type S in week $n$''; $M_{n}$: ``the individual is sick in week $n$''; $I_{n}$: ``the individual is immunized in week $n$''. In week 0, all individuals are considered ``of type S'', so: $$P(S_{0}) = 1 ; \quad P(M_{0}) = 0 \quad \text{and} \quad P(I_{0}) = 0.$$

Part A

We study the evolution of the epidemic during weeks 1 and 2.

1. Reproduce and complete the probability tree.
2. Show that $P(I_{2}) = 0.2025$.
3. Given that an individual is immunized in week 2, what is the probability, rounded to the nearest thousandth, that he was sick in week 1?

Part B

We study the long-term evolution of the disease. For any natural integer $n$, we have: $u_{n} = P(S_{n})$, $v_{n} = P(M_{n})$ and $w_{n} = P(I_{n})$.

A customer is chosen at random from those who bought a melon during week 1. Among customers who buy a melon in a given week, $90\%$ of them buy a melon the following week; among customers who do not buy a melon in a given week, $60\%$ of them do not buy a melon the following week. For $n \geqslant 1$, we set $p_n = P(A_n)$, where $A_n$ is the event ``the customer buys a melon during week $n$''. Thus $p_1 = 1$. Prove that, for all integer $n \geqslant 1$: $p_{n+1} = 0{,}5\, p_n + 0{,}4$.

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?

A bag contains the following eight letters: A B C D E F G H (2 vowels and 6 consonants).
A game consists of drawing simultaneously at random two letters from this bag. You win if the draw consists of one vowel and one consonant.

1. A player draws simultaneously two letters from the bag. a. Determine the number of possible draws. b. Determine the probability that the player wins this game.

Questions 2 and 3 of this exercise are independent.

For the rest of the exercise, we admit that the probability that the player wins is equal to $\frac{3}{7}$.

1. To play, the player must pay $k$ euros, where $k$ is a non-zero natural integer. If the player wins, he receives 10 euros, otherwise he receives nothing. We denote $G$ the random variable equal to the algebraic gain of a player (that is, the sum received minus the sum paid). a. Determine the probability distribution of $G$. b. What must be the maximum value of the sum paid at the start for the game to remain favourable to the player?
2. Ten players each play one game. The letters drawn are returned to the bag after each game. We denote $X$ the random variable equal to the number of winning players. a. Justify that $X$ follows a binomial distribution and give its parameters. b. Calculate the probability, rounded to $10^{-3}$, that there are exactly four winning players. c. Calculate $P(X \geqslant 5)$ by rounding to $10^{-3}$. Give an interpretation of the result obtained. d. Determine the smallest natural integer $n$ such that $P(X \leqslant n) \geqslant 0.9$.

Exercise 1 — 7 points

Topics: Probability

In Hugo's shop, customers can rent two types of bicycles: road bikes or mountain bikes. Each type of bicycle can be rented in an electric version or not.
A customer is chosen at random from the shop, and we assume that:

• If the customer rents a road bike, the probability that it is an electric bike is 0.4;
• If the customer rents a mountain bike, the probability that it is an electric bike is 0.7;
• The probability that the customer rents an electric bike is 0.58.

We denote by $\alpha$ the probability that the customer rents a road bike, with $0 \leqslant \alpha \leqslant 1$. We consider the following events:

• R: ``the customer rents a road bike'';
• $E$ : ``the customer rents an electric bike'';
• $\bar { R }$ and $\bar { E }$, complementary events of $R$ and $E$.

We model this random situation using the tree shown below. If $F$ denotes any event, we denote by $p ( F )$ the probability of $F$.

1. Copy this tree onto your answer sheet and complete it.
2. a. Show that $p ( E ) = 0.7 - 0.3 \alpha$. b. Deduce that: $\alpha = 0.4$.
3. We know that the customer rented an electric bike. Determine the probability that they rented a mountain bike. Give the result rounded to the nearest hundredth.
4. What is the probability that the customer rents an electric mountain bike?
5. The daily rental price of a non-electric road bike is 25 euros, that of a non-electric mountain bike is 35 euros. For each type of bike, choosing the electric version increases the daily rental price by 15 euros. We denote by $X$ the random variable modeling the daily rental price of a bike. a. Give the probability distribution of $X$. Present the results in the form of a table. b. Calculate the expected value of $X$ and interpret this result.
6. When 30 of Hugo's customers are chosen at random, we treat this choice as sampling with replacement. We denote by $Y$ the random variable associating to a sample of 30 randomly chosen customers the number of customers who rent an electric bike. We recall that the probability of event $E$ is: $p ( E ) = 0.58$. a. Justify that $Y$ follows a binomial distribution and specify its parameters. b. Determine the probability that a sample contains exactly 20 customers who rent an electric bike. Give the result rounded to the nearest thousandth. c. Determine the probability that a sample contains at least 15 customers who rent an electric bike. Give the result rounded to the nearest thousandth.

Exercise 4 — 7 points Theme: Probability An urn contains white and black tokens all indistinguishable to the touch.
A game consists of drawing at random successively and with replacement two tokens from this urn. The following game rule is established:

• a player loses 9 euros if the two tokens drawn are white;
• a player loses 1 euro if the two tokens drawn are black;
• a player wins 5 euros if the two tokens drawn are of different colors.

1. We consider that the urn contains 2 black tokens and 3 white tokens.
   1. [a.] Model the situation using a probability tree.
   2. [b.] Calculate the probability of losing $9\,\text{\euro}$ in one game.
   
   
2. We now consider that the urn contains 3 white tokens and at least two black tokens but we do not know the exact number of black tokens. We will call $N$ the number of black tokens.
   1. [a.] Let $X$ be the random variable giving the gain of the game for one game. Determine the probability distribution of this random variable.
   2. [b.] Solve the inequality for real $x$: $$-x^2 + 30x - 81 > 0$$
   3. [c.] Using the result of the previous question, determine the number of black tokens the urn must contain so that this game is favorable to the player.
   4. [d.] How many black tokens should the player request in order to obtain a maximum average gain?
   
   
3. We observe 10 players who try their luck by playing one game of this game, independently of each other. We assume that 7 black tokens have been placed in the urn (with 3 white tokens). What is the probability of having at least 1 player winning 5 euros?

A production company is considering whether to schedule a television game show. This game brings together four candidates and takes place in two phases:

• The first phase is a qualification phase. This phase depends only on chance. For each candidate, the probability of qualifying is 0.6.
• The second phase is a competition between the qualified candidates. It only takes place if at least two candidates are qualified. Its duration depends on the number of qualified candidates as indicated in the table below (when there is no second phase, its duration is considered to be zero).

\begin{tabular}{ l } Number of candidates qualified

for the second phase

& 0 & 1 & 2 & 3 & 4 \hline

Duration of the second phase in

minutes

& 0 & 0 & 5 & 9 & 11 \hline \end{tabular}
For the company to decide to retain this game, the following two conditions must be verified: Condition no. 1: The second phase must take place in at least 80\% of cases. Condition no. 2: The average duration of the second phase must not exceed 6 minutes. Can the game be retained?

This exercise is a multiple choice questionnaire. For each question, only one of the four proposed answers is correct. No justification is required. A wrong answer, a multiple answer, or the absence of an answer to a question neither awards nor deducts points. The five questions are independent.

1. We consider the function $f$ defined on the interval $]1; +\infty[$ by $$f(x) = 0{,}05 - \frac{\ln x}{x-1}$$ The limit of the function $f$ at $+\infty$ is equal to: a. $+\infty$ b. 0.05 c. $-\infty$ d. 0
   
2. We consider a function $h$ continuous on the interval $[-2;4]$ such that: $$h(-1) = 0, \quad h(1) = 4, \quad h(3) = -1.$$ We can affirm that: a. the function $h$ is increasing on the interval $[-1; 1]$. b. the function $h$ is positive on the interval $[-1; 1]$. c. there exists at least one real number $a$ in the interval $[1; 3]$ such that $h(a) = 1$. d. the equation $h(x) = 1$ has exactly two solutions in the interval $[-2; 4]$.
   
3. We consider two sequences $(u_{n})$ and $(v_{n})$ with strictly positive terms such that $\lim_{n \rightarrow +\infty} u_{n} = +\infty$ and $(v_{n})$ converges to 0. We can affirm that: a. the sequence $\left(\dfrac{1}{v_{n}}\right)$ converges. b. the sequence $\left(\dfrac{v_{n}}{u_{n}}\right)$ converges. c. the sequence $(u_{n})$ is increasing. d. $\lim_{n \rightarrow +\infty} \left(-u_{n}\right)^{n} = -\infty$
   
4. To participate in a game, a player must pay $4\,€$. They then roll a fair six-sided die:
   • if they get 1, they win $12\,€$;
   • if they get an even number, they win $3\,€$;
   • otherwise, they win nothing.
   On average, the player: a. wins $3.50\,€$ b. loses $3\,€$. c. loses $1.50\,€$ d. loses $0.50\,€$.
   
5. We consider the random variable $X$ following the binomial distribution $\mathscr{B}(3; p)$. We know that $P(X = 0) = \dfrac{1}{125}$. We can affirm that: a. $p = \dfrac{1}{5}$ b. $P(X = 1) = \dfrac{124}{125}$ c. $p = \dfrac{4}{5}$ d. $P(X = 1) = \dfrac{4}{5}$

For each of the five questions in this exercise, only one of the four proposed answers is correct. The candidate will indicate on his/her work the number of the question and the chosen answer. No justification is required.
A wrong answer, multiple answers, or the absence of an answer to a question neither gives nor removes points.
An urn contains 15 indistinguishable balls to the touch, numbered from 1 to 15. The ball numbered 1 is red. The balls numbered 2 to 5 are blue. The other balls are green. We choose a ball at random from the urn. We denote $R$ (respectively $B$ and $V$) the event: ``The ball drawn is red'' (respectively blue and green).
Question 1: What is the probability that the ball drawn is blue or numbered with an even number?

Answer A	Answer B	Answer C	Answer D
$\frac { 7 } { 15 }$	$\frac { 9 } { 15 }$	$\frac { 11 } { 10 }$	None of the previous statements is correct.

Question 2: Given that the ball drawn is green, what is the probability that it is numbered 7?

Answer A	Answer B	Answer C	Answer D
$\frac { 1 } { 15 }$	$\frac { 7 } { 15 }$	$\frac { 1 } { 10 }$	None of the previous statements is correct.

A game is set up. To be able to play, the player pays the sum of 10 euros called the stake. This game consists of drawing a ball at random from the urn.

• If the ball drawn is blue, the player wins, in euros, three times the number of the ball.
• If the ball drawn is green, the player wins, in euros, the number of the ball.
• If the ball drawn is red, the player wins nothing.

We denote $G$ the random variable that gives the algebraic gain of the player, that is, the difference between what he wins and his initial stake. For example, if the player draws the blue ball numbered 3, then his algebraic gain is $-1$ euro.
Question 3: What is the value of $P ( G = 5 )$ ?

Answer A	Answer B	Answer C	Answer D
$\frac { 1 } { 15 }$	$\frac { 2 } { 15 }$	$\frac { 1 } { 3 }$	None of the previous statements is correct.

Question 4: What is the value of $P _ { R } ( G = 0 )$ ?

Answer A	Answer B	Answer C	Answer D
0	$\frac { 1 } { 15 }$	1	None of the previous statements is correct.

Question 5: What is the value of $P _ { ( G = - 4 ) } ( V )$ ?

Answer A	Answer B	Answer C	Answer D
$\frac { 1 } { 15 }$	$\frac { 4 } { 15 }$	$\frac { 1 } { 2 }$	None of the previous statements is correct.

An opaque bag contains eight tokens numbered from 1 to 8, indistinguishable to the touch. Three times, a player draws a token from this bag, notes its number, then puts it back in the bag. In this context, we call a ``draw'' the ordered list of the three numbers obtained. For example, if the player draws token number 4, then token number 5, then token number 1, then the corresponding draw is $(4 ; 5 ; 1)$.

1. Determine the number of possible draws.
2. 1. [a.] Determine the number of draws without repetition of numbers.
   2. [b.] Deduce from this the number of draws containing at least one repetition of numbers.

We denote $X_1$ the random variable equal to the number of the first token drawn, $X_2$ the one equal to the number of the second token drawn and $X_3$ the one equal to the number of the third token drawn. Since this is a draw with replacement, the random variables $X_1, X_2$, and $X_3$ are independent and follow the same probability distribution.

1. Establish the probability distribution of the random variable $X_1$.
2. Determine the expectation of the random variable $X_1$.

We denote $S = X_1 + X_2 + X_3$ the random variable equal to the sum of the numbers of the three tokens drawn.

1. Determine the expectation of the random variable $S$.
2. Determine $P(S = 24)$.
3. If a player obtains a sum greater than or equal to 22, then they win a prize.
   1. [a.] Justify that there are exactly 10 draws allowing one to win a prize.
   2. [b.] Deduce from this the probability of winning a prize.

Exercise 4
For each of the following statements, indicate whether it is true or false. Each answer must be justified. An unjustified answer earns no points.
A museum offers visits with or without an audioguide. Tickets can be purchased online or directly at the counter.

1. When a person buys their ticket online, a validation code is sent to them by SMS so they can confirm their purchase. This code is generated randomly and consists of 4 digits that are pairwise distinct, with the first digit being different from 0.
   Statement 1: The number of different codes that can be generated is 5040.
   
2. A study made it possible to consider that:
   • the probability that a person chooses the audioguide given that they bought their ticket online is equal to 0{,}8;
   • the probability that a person buys their ticket online is equal to 0{,}7;
   • the probability that a person opts for a visit without an audioguide is equal to 0{,}32.
   
   Statement 2: The probability that a visitor does not take the audioguide given that they bought their ticket at the counter is greater than two thirds.
   
3. We randomly choose 12 visitors to this museum.
   We assume that the choice of the ``audioguide'' option is independent from one visitor to another.
   Statement 3: The probability that exactly half of these visitors opt for the audioguide is equal to $924 \times 0{,}2176^6$.
   
4. When a person has an audioguide, they can choose from three routes:
   • a first one lasting fifty minutes,
   • a second one lasting one hour and twenty minutes,
   • a third one lasting one hour and forty minutes.
   
   The tour time can be modelled by a random variable $X$ whose probability distribution is given below:
   

   $x_i$	$50\,\min$	$1\,\mathrm{h}\,20\,\min$	$1\,\mathrm{h}\,40\,\min$
   $P(X = x_i)$	0{,}1	0{,}6	0{,}3

   
   Statement 4: The expectation of $X$ is 77 minutes.

Three cubic dice, with faces numbered from 1 to 6, were used in a game. Artur chose two dice, and João got the third. The game consists of both rolling their dice, observing the numbers on the faces facing up, and comparing the largest number obtained by Artur with the number obtained by João. The player who obtains the largest number wins. In case of a tie, the victory goes to João.
The player who has the greatest probability of victory is
(A) Artur, with probability of $\dfrac{2}{3}$
(B) João, with probability of $\dfrac{4}{9}$
(C) Artur, with probability of $\dfrac{91}{216}$
(D) João, with probability of $\dfrac{91}{216}$
(E) Artur, with probability of $\dfrac{125}{216}$

(a) $n$ identical chocolates are to be distributed among the $k$ students in Tinku's class. Find the probability that Tinku gets at least one chocolate, assuming that the $n$ chocolates are handed out one by one in $n$ independent steps. At each step, one chocolate is given to a randomly chosen student, with each student having equal chance to receive it.
(b) Solve the same problem assuming instead that all distributions are equally likely. You are given that the number of such distributions is $\binom { n + k - 1 } { k - 1 }$. (Here all chocolates are considered interchangeable but students are considered different.)

Ten people sitting around a circular table decide to donate some money for charity. You are told that the amount donated by each person was the average of the money donated by the two persons sitting adjacent to him/her. One person donated Rs. 500. Choose the correct option for each of the following two questions. Write your answers as a sequence of two letters (a/b/c/d).
What is the total amount donated by the 10 people?
(a) exactly Rs. 5000
(b) less than Rs. 5000
(c) more than Rs. 5000
(d) not possible to decide among the above three options.
What is the maximum amount donated by an individual?
(a) exactly Rs. 500
(b) less than Rs. 500
(c) more than Rs. 500
(d) not possible to decide among the above three options.

The probability distribution table of random variable $X$ is shown below. Find the variance of random variable $Y = 10 X + 5$. [3 points]

$X$	0	1	2	3	Total
$\mathrm { P } ( X )$	$\frac { 2 } { 10 }$	$\frac { 3 } { 10 }$	$\frac { 3 } { 10 }$	$\frac { 2 } { 10 }$	1

A discrete random variable $X$ can take values $0,1,2,3,4,5,6,7$ and its probability mass function is $$\mathrm { P } ( X = x ) = \left\{ \begin{array} { l l } c , & x = 0,1,2 \\ 2 c , & x = 3,4,5 \\ 5 c ^ { 2 } , & x = 6,7 \end{array} \quad ( \text { where } c \text { is a positive number } ) \right.$$ Let $A$ be the event that the random variable $X$ is at least 6, and let $B$ be the event that the random variable $X$ is at least 3. What is the value of $\mathrm { P } ( A \mid B )$? [3 points]
(1) $\frac { 1 } { 5 }$
(2) $\frac { 1 } { 6 }$
(3) $\frac { 1 } { 7 }$
(4) $\frac { 1 } { 8 }$
(5) $\frac { 1 } { 9 }$

The following is a probability distribution table for the random variable $X$.

$X$	$k$	$2 k$	$4 k$	Total
$\mathrm { P } ( X = x )$	$\frac { 4 } { 7 }$	$a$	$b$	1

If $\frac { 4 } { 7 } , a , b$ form a geometric sequence in this order and the mean of $X$ is 24, find the value of $k$. [3 points]

There is a regular tetrahedron-shaped box with the numbers $1,1,1,2$ written one on each face. When this box is thrown, if the number on the bottom face is 1, region A in the figure on the right is colored, and if the number is 2, region B is colored. When the box is thrown repeatedly until both regions are colored, find the probability that the process is completed on the 3rd throw. If this probability is $\frac { q } { p }$, find the value of $p + q$. (Here, $p , q$ are coprime natural numbers.) [4 points]

There is a regular tetrahedron-shaped box with the numbers $1,1,1,2$ written one on each face. When this box is thrown, if the number on the bottom face is 1, color region A in the figure on the right; if the number is 2, color region B. Continue throwing this box until both regions are colored. Find the probability that the process is completed on the 3rd throw, expressed as $\frac { q } { p }$. Find the value of $p + q$. (Here, $p , q$ are coprime natural numbers.) [4 points]

A factory produces products that are sold with 50 items per box. The number of defective items in a box follows a binomial distribution with mean $m$ and variance $\frac { 48 } { 25 }$. Before selling a box, all 50 products are inspected to find defective items, which costs a total of 60,000 won. If a box is sold without inspection, an after-sales service cost of $a$ won is required for each defective item.
When the expected value of the cost of inspecting all products in a box equals the expected cost of after-sales service, find the value of $\frac { a } { 1000 }$. (Given that $a$ is a constant and $m$ is a natural number not exceeding 5.) [4 points]

(Probability and Statistics) A coin is tossed three times, and based on the results, a score is obtained as a random variable $X$ according to the following rules. (가) If the same face does not appear consecutively, the score is 0 points. (나) If the same face appears consecutively exactly twice, the score is 1 point. (다) If the same face appears consecutively three times, the score is 3 points.
What is the variance $\mathrm{V}(X)$ of the random variable $X$? [3 points]
(1) $\frac{9}{8}$
(2) $\frac{19}{16}$
(3) $\frac{5}{4}$
(4) $\frac{21}{16}$
(5) $\frac{11}{8}$

The probability distribution table of the random variable $X$ is as follows.

$X$	0	1	2	Total
$\mathrm { P } ( X = x )$	$\frac { 2 } { 7 }$	$\frac { 3 } { 7 }$	$\frac { 2 } { 7 }$	1

What is the value of the variance $\mathrm { V } ( 7 X )$ of the random variable $7 X$? [3 points]
(1) 14
(2) 21
(3) 28
(4) 35
(5) 42

The probability distribution table of the random variable $X$ is as follows.

$X$	- 1	0	1	2	Total
$\mathrm { P } ( X = x )$	$\frac { 3 - a } { 8 }$	$\frac { 1 } { 8 }$	$\frac { 3 + a } { 8 }$	$\frac { 1 } { 8 }$	1

When $\mathrm { P } ( 0 \leqq X \leqq 2 ) = \frac { 7 } { 8 }$, what is the value of the expected value $\mathrm { E } ( X )$ of the random variable $X$? [3 points]
(1) $\frac { 1 } { 4 }$
(2) $\frac { 3 } { 8 }$
(3) $\frac { 1 } { 2 }$
(4) $\frac { 5 } { 8 }$
(5) $\frac { 3 } { 4 }$