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 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?

There are 5 drawers, each labeled with a natural number from 1 to 5. Two drawers are randomly assigned to Younghee. Let $X$ be the random variable representing the smaller of the two natural numbers on the assigned drawers. Find the value of $\mathrm { E } ( 10 X )$. [4 points]

Point P is at the origin of the coordinate plane. The following trial is performed using one die.
When the die is rolled and the number shown is
2 or less, point P is moved 3 units in the positive direction of the $x$-axis,
3 or more, point P is moved 1 unit in the positive direction of the $y$-axis.
This trial is repeated 15 times, and the distance between the moved point P and the line $3 x + 4 y = 0$ is the random variable $X$. What is the value of $\mathrm { E } ( X )$? [4 points]
(1) 13
(2) 15
(3) 17
(4) 19
(5) 21

When 4 coins are tossed simultaneously, let $X$ be the random variable representing the number of coins showing heads. Define the discrete random variable $Y$ as $$Y = \begin{cases} X & (\text{if } X \text{ takes the value } 0 \text{ or } 1) \\ 2 & (\text{if } X \text{ takes a value of } 2 \text{ or more}) \end{cases}$$ Find the value of $\mathrm{E}(Y)$. [3 points]
(1) $\frac{25}{16}$
(2) $\frac{13}{8}$
(3) $\frac{27}{16}$
(4) $\frac{7}{4}$
(5) $\frac{29}{16}$

16. (This question is worth 13 points) Groups $A$ and $B$ each have 7 patients. Their recovery time (in days) after taking a certain drug is recorded as follows: Group A: $10,11,12,13,14,15,16$ Group B: $12,13,15,16,17,14 , a$ Assume that the recovery times of all patients are mutually independent. Randomly select 1 person from each of groups A and B. The person selected from group A is denoted as patient 甲, and the person selected from group B is denoted as patient 乙. (I) Find the probability that the recovery time of patient 甲 is at least 14 days; (II) If $a = 25$, find the probability that the recovery time of patient 甲 is longer than that of patient 乙; (III) For what value of $a$ are the variances of recovery times for groups A and B equal? (Proof of the conclusion is not required)

A factory accepted a processing contract. The processed products (unit: pieces) are classified into four grades: A, B, C, and D according to standards. According to the contract: for grade A, B, and C products, the customer pays processing fees of 90 yuan, 50 yuan, and 20 yuan per piece respectively; for grade D products, the factory must compensate 50 yuan per piece for raw material loss. The factory has two branch factories, Factory A and Factory B, that can undertake the processing contract. Factory A has a processing cost of 25 yuan per piece, and Factory B has a processing cost of 20 yuan per piece. To decide which branch factory should undertake the contract, the factory conducted trial processing of 100 pieces of this product at each branch factory and recorded the grades of these products, as shown below:
Frequency Distribution Table of Product Grades for Factory A:

Grade	A	B	C	D
Frequency	40	20	20	20

Frequency Distribution Table of Product Grades for Factory B:

Grade	A	B	C	D
Frequency	28	17	34	21

(1) Estimate the probability that a product from Factory A and Factory B respectively is grade A;
(2) Find the average profit for 100 products from Factory A and Factory B respectively. Based on average profit, which branch factory should the factory choose to undertake the contract?

Given that the premium for a certain insurance is 0.4 ten thousand yuan. For the first 3 claims, each claim pays 0.8 ten thousand yuan; the 4th claim pays 0.6 ten thousand yuan.

Number of Claims	0	1	2	3	4
Number of Policies	800	100	60	30	10

A sample of 100 policies is drawn from the population. Using frequency to estimate probability:
(1) Find the probability that a randomly selected policy has at least 2 claims;
(2) (i) Gross profit is the difference between premium and claim amount. Let gross profit be $X$. Estimate the mathematical expectation of $X$;
(ii) If policies with no claims have their premium reduced by 4\% in the next insurance period, and policies with claims have their premium increased by 20\%, estimate the mathematical expectation of gross profit for the next insurance period.

Let $\left(X_{n}\right)_{n \in \mathbb{N}}$ be a sequence of mutually independent Rademacher random variables and $S_{n} = \sum_{j=1}^{n} X_{j}$. For $x$ real, $\lfloor x \rfloor$ denotes the integer part of $x$. For all real numbers $\delta > 0$ and $\tau > 0$, calculate $\mathbb{V}\left(\delta S_{\lfloor 1/\tau \rfloor}\right)$, the variance of the random variable $\delta S_{\lfloor 1/\tau \rfloor}$.

We consider an urn containing $A$ balls of which $pA$ are white and $qA$ are black. We draw simultaneously $n$ balls from the urn. We number from 1 to $pA$ each of the white balls and, for any natural integer $i \in \llbracket 1, pA \rrbracket$, we define $$Y_i = \begin{cases} 1 & \text{if the ball numbered } i \text{ was drawn,} \\ 0 & \text{otherwise.} \end{cases}$$ Let $Y$ denote the number of white balls drawn. Express $Y$ using the $Y_i$ and recover the value of the expectation of $Y$. Compare it to that of $Z$ (where $Z \sim \mathcal{B}(n,p)$ is the number of white balls in $n$ draws with replacement).

Consider the $6 \times 6$ square in the figure. Let $A _ { 1 } , A _ { 2 } , \ldots , A _ { 49 }$ be the points of intersections (dots in the picture) in some order. We say that $A _ { i }$ and $A _ { j }$ are friends if they are adjacent along a row or along a column. Assume that each point $A _ { i }$ has an equal chance of being chosen.
Let $p _ { i }$ be the probability that a randomly chosen point has $i$ many friends, $i = 0,1,2,3,4$. Let $X$ be a random variable such that for $i = 0,1,2,3,4$, the probability $P ( X = i ) = p _ { i }$. Then the value of $7 E ( X )$ is

An unbiased coin is tossed 5 times. Suppose that a variable $X$ is assigned the value $k$ when $k$ consecutive heads are obtained for $k = 3, 4, 5$, otherwise $X$ takes the value $-1$. Then the expected value of $X$, is
(1) $\frac { 3 } { 16 }$
(2) $\frac { 1 } { 8 }$
(3) $- \frac { 3 } { 16 }$
(4) $- \frac { 1 } { 8 }$

Three balls are drawn at random from a bag containing 5 blue and 4 yellow balls. Let the random variables $X$ and $Y$ respectively denote the number of blue and yellow balls. If $\bar { X }$ and $\bar { Y }$ are the means of $X$ and $Y$ respectively, then $7 \bar { X } + 4 \bar { Y }$ is equal to $\_\_\_\_$

A coin is tossed three times. Let $X$ denote the number of times a tail follows a head. If $\mu$ and $\sigma ^ { 2 }$ denote the mean and variance of $X$, then the value of $64 \left( \mu + \sigma ^ { 2 } \right)$ is:
(1) 51
(2) 64
(3) 32
(4) 48

In a bag there are a total of nine balls: one white, three red and five black. The white ball is worth five points, a red ball is worth three points, and a black ball is worth one point. Two balls are taken from the bag together, and the trial is scored by the sum of the values of the two balls.
(1) The highest possible score is $\mathbf{A}$, and the probability that it happens is $\dfrac{\mathbf{B}}{\mathbf{CD}}$.
(2) The probability that the score is 6 is $\dfrac{\mathbf{E}}{\mathbf{F}}$.
(3) The expected value of the score is $\dfrac{\mathbf{GH}}{\mathbf{I}}$.

There are two boxes $A$ and $B$. Box $A$ contains 6 white balls and 4 red balls, and box $B$ contains 8 white balls and 2 blue balls. There are three lottery methods (each ball in each box has equal probability of being drawn): (I) First draw one ball from box $A$; if a red ball is drawn, stop; if a white ball is drawn, then draw one ball from box $B$; (II) First draw one ball from box $B$; if a blue ball is drawn, stop; if a white ball is drawn, then draw one ball from box $A$; (III) Simultaneously draw one ball from each of boxes $A$ and $B$. The prize rules are: Among red and blue balls, if only a red ball is drawn, win 50 yuan; if only a blue ball is drawn, win 100 yuan; if both colors are drawn, still win only 100 yuan; if neither color is drawn, win nothing. Let $E_{1}$, $E_{2}$, $E_{3}$ denote the expected values of winnings for methods (I), (II), (III) respectively. Select the correct option.
(1) $E_{1} > E_{2} > E_{3}$
(2) $E_{1} = E_{2} > E_{3}$
(3) $E_{2} = E_{3} > E_{1}$
(4) $E_{1} = E_{3} > E_{2}$
(5) $E_{3} > E_{2} > E_{1}$

There is a game involving moving a game piece on a number line. The way to move the piece is determined by rolling a fair die, with the following rules: (I) When the die shows 1 point, the piece does not move. (II) When the die shows 3 or 5 points, the piece moves left (negative direction) by ``that point number minus 1'' units. (III) When the die shows an even number, the piece moves right (positive direction) by ``half of that point number'' units. On the first die roll, the piece starts at the origin. From the second roll onwards, the piece starts from the position it was in after the previous roll. For example, if two die rolls result in 5 points and 2 points respectively, the piece first moves left 4 units to coordinate $-4$, then moves right 1 unit to coordinate $-3$. Select the correct options.
(1) After rolling the die once, the probability that the piece is at distance 2 from the origin is $\frac { 1 } { 2 }$
(2) After rolling the die once, the expected value of the piece's coordinate is 0
(3) After rolling the die twice, the piece's coordinate could be $-5$
(4) After rolling the die twice, given that the sum of the two rolls is odd, the probability that the piece's coordinate is positive is $\frac { 4 } { 9 }$
(5) After rolling the die three times, the probability that the piece is at the origin is $\left( \frac { 1 } { 6 } \right) ^ { 3 }$

A company holds a year-end lottery. Each person randomly draws two cards from six cards numbered 1 to 6. Assume each card has an equal chance of being drawn, and the rules are as follows: (I) If the sum of the numbers on the two cards is odd, the person wins 100 yuan and the lottery ends; (II) If the sum is even, the two cards are discarded, and two cards are randomly drawn from the remaining four cards. If the sum of their numbers is odd, the person wins 50 yuan; otherwise, there is no prize and the lottery ends. According to the above rules, what is the expected value of the prize money for each person participating in this lottery?
(1) 50
(2) 70
(3) 72
(4) 80
(5) 100

There is a card-drawing prize activity with the following rules: In an opaque box, there are 2 cards marked with ``1000 yuan'' and 3 cards marked with ``0 yuan''. A participant randomly draws one card from the box. Without knowing the amount marked on the drawn card, the host then places a card marked with ``500 yuan'' into the box. At this point, the participant has two choices: (I) Keep the originally drawn card; the amount marked on that card is the prize won. (II) Discard the originally drawn card without returning it, and randomly draw another card from the box; the amount marked on that card is the prize won.
A participant joins this activity. Assume each card has an equal chance of being drawn. Select the correct options.
(1) If the participant chooses (I), the probability of winning 0 yuan is $\frac { 3 } { 5 }$
(2) If the participant chooses (I), the expected value of the prize is 500 yuan
(3) If the participant chooses (II), the probability of winning 1000 yuan is $\frac { 2 } { 5 }$
(4) If the participant chooses (II), the probability of winning 0 yuan is $\frac { 12 } { 25 }$
(5) If the participant chooses (II), the expected value of the prize is 420 yuan

A store sells a popular action figure through a lottery. Each lottery draw is independent with a probability of winning of $\frac{2}{5}$. Participants can participate in the lottery using one of the following two methods.
Method 1: Pay 225 yuan to get two lottery chances. Stop drawing as soon as you win and receive one action figure. If you fail to win in both draws, you must pay an additional 75 yuan to receive one action figure.
Method 2: Unlimited number of lottery draws, paying 100 yuan per draw.
Assuming there is no limit on spending until obtaining one action figure, find the expected value of the amount paid to obtain one action figure using each of the two lottery methods, and explain the relationship between these two expected values. (Non-multiple choice question, 6 points)

A holiday market stall offers ``test your luck—cute dolls regularly priced at 480 yuan can be purchased for as low as 240 yuan''. The rules are: customers flip a fair coin up to 5 times. If 3 consecutive heads are obtained in the first 3 flips, they can purchase a doll for 240 yuan. If 3 heads are accumulated by the 4th flip, they can purchase for 320 yuan. If 3 heads are accumulated by the 5th flip, they can purchase for 400 yuan. If 3 heads are not accumulated after 5 flips, they can purchase for 480 yuan. The expected value of the amount a customer spends to purchase a doll is (15-1) (15-2) (15-3) yuan.