Part A - First model
Based on a data sample, we consider an initial modelling:

• each year, the probability that the El Niño phenomenon is dominant is equal to 0.4;
• the occurrence of the El Niño phenomenon occurs independently from one year to the next.

We denote by $X$ the random variable which, over a period of 10 years, associates the number of years in which El Niño is dominant.

1. Justify that $X$ follows a binomial distribution and specify the parameters of this distribution.
2. a. Calculate the probability that, over a period of 10 years, the El Niño phenomenon is dominant in exactly 2 years. b. Calculate $P ( X \leqslant 2 )$. What does this result mean in the context of the exercise?
3. Calculate $E ( X )$. Interpret this result.

Dominique answers a multiple choice questionnaire with 10 questions. For each question, 4 answers are proposed, of which only one is correct. Dominique answers randomly to each of the 10 questions by checking, for each question, exactly one box among the 4. For each question, the probability that he answers correctly is therefore $\frac { 1 } { 4 }$. We denote $X$ the random variable that counts the number of correct answers to this questionnaire.

1. Determine the distribution followed by the random variable $X$ and give the parameters of this distribution.
2. What is the probability that Dominique obtains exactly 5 correct answers? Round the result to $10 ^ { - 4 }$ near.
3. Give the expectation of $X$ and interpret this result in the context of the exercise.
4. We suppose in this question that a correct answer gives one point and an incorrect answer loses 0.5 points. The final score can therefore be negative.

We denote $Y$ the random variable that gives the number of points obtained. a. Calculate $P ( Y = 10 )$, give the exact value of the result. b. From how many correct answers is Dominique's final score positive? Justify. c. Calculate $P ( Y \leqslant 0 )$, give an approximate value to the nearest hundredth. d. Show that $Y = 1.5 X - 5$. e. Calculate the expectation of the random variable $Y$.

5. During a blood collection, a sample of 100 people is chosen from the population of a French city. This population is large enough to assimilate this choice to sampling with replacement. We denote by $X$ the random variable that associates to each sample of 100 people the number of universal donors in that sample. a. Justify that $X$ follows a binomial distribution and specify its parameters. b. Determine to $10 ^ { - 3 }$ near the probability that there are at most 7 universal donors in this sample. c. Show that the expectation $E ( X )$ of the random variable $X$ is equal to 7.14 and that its variance $V ( X )$ is equal to 6.63 to $10 ^ { - 2 }$ near.

On an avenue there are 10 traffic lights. Due to a system failure, the traffic lights were without control for one hour, and fixed their lights only in green or red. The traffic lights operate independently; the probability of showing green is $\frac{2}{3}$ and of showing red is $\frac{1}{3}$. A person walked the entire avenue during the period of the failure, observing the color of the light of each of these traffic lights. What is the probability that this person observed exactly one signal in green?
(A) $\frac{10 \times 2}{3^{10}}$
(B) $\frac{10 \times 2^{9}}{3^{10}}$
(C) $\frac{2^{10}}{3^{100}}$
(D) $\frac{2^{90}}{3^{100}}$
(E) $\frac{2}{3^{10}}$

When a random variable $X$ follows a binomial distribution $\mathrm { B } \left( 100 , \frac { 1 } { 5 } \right)$, what is the standard deviation of the random variable $3 X - 4$? [3 points]
(1) 12
(2) 15
(3) 18
(4) 21
(5) 24

There is a television with channels set from 1 to 100. The currently active channel is 50. When one button is randomly pressed six times, either the channel increase button or the channel decrease button, what is the probability that the channel returns to 50? (Note: Each time a button is pressed, the channel changes by 1.) [4 points]
(1) $\frac { 1 } { 4 }$
(2) $\frac { 5 } { 16 }$
(3) $\frac { 3 } { 8 }$
(4) $\frac { 7 } { 16 }$
(5) $\frac { 1 } { 2 }$

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]

When rolling a die 20 times, let $X$ be the random variable representing the number of times the face 1 appears, and when tossing a coin $n$ times, let $Y$ be the random variable representing the number of times heads appears. Find the minimum value of $n$ such that the variance of $Y$ is greater than the variance of $X$. [4 points]

When the trial of simultaneously tossing 2 coins is repeated 10 times, let $X$ be the random variable representing the number of times both coins show heads. Find the variance $\mathrm { V } ( 4 X + 1 )$ of the random variable $4 X + 1$. [3 points]

A random variable $X$ follows a binomial distribution $\mathrm { B } ( 200 , p )$ and the mean of $X$ is 40. What is the variance of $X$? [2 points]
(1) 32
(2) 33
(3) 34
(4) 35
(5) 36

A random variable $X$ follows a binomial distribution $\mathrm{B}(n, p)$. If the mean and standard deviation of the random variable $2X - 5$ are 175 and 12, respectively, what is the value of $n$? [3 points]
(1) 130
(2) 135
(3) 140
(4) 145
(5) 150

A random variable $X$ follows a binomial distribution $\mathrm { B } ( 9 , p )$, and $\{ \mathrm { E } ( X ) \} ^ { 2 } = \mathrm { V } ( X )$. What is the value of $p$? (Here, $0 < p < 1$) [3 points]
(1) $\frac { 1 } { 13 }$
(2) $\frac { 1 } { 12 }$
(3) $\frac { 1 } { 11 }$
(4) $\frac { 1 } { 10 }$
(5) $\frac { 1 } { 9 }$

A random variable $X$ follows a binomial distribution $\mathrm { B } \left( n , \frac { 1 } { 3 } \right)$ and $\mathrm { V} ( 3 X ) = 40$. Find the value of $n$. [3 points]

When a coin is tossed 5 times, what is the probability that the product of the number of heads and the number of tails is 6? [3 points]
(1) $\frac { 5 } { 8 }$
(2) $\frac { 9 } { 16 }$
(3) $\frac { 1 } { 2 }$
(4) $\frac { 7 } { 16 }$
(5) $\frac { 3 } { 8 }$

When rolling a die three times, what is the probability that the number 4 appears exactly once? [3 points]
(1) $\frac { 25 } { 72 }$
(2) $\frac { 13 } { 36 }$
(3) $\frac { 3 } { 8 }$
(4) $\frac { 7 } { 18 }$
(5) $\frac { 29 } { 72 }$

When rolling a die 3 times, what is the probability that the number 4 appears exactly once? [3 points]
(1) $\frac { 25 } { 72 }$
(2) $\frac { 13 } { 36 }$
(3) $\frac { 3 } { 8 }$
(4) $\frac { 7 } { 18 }$
(5) $\frac { 29 } { 72 }$

When a coin is tossed 6 times, the probability that the number of heads is greater than the number of tails is $\frac { q } { p }$. Find the value of $p + q$. (Here, $p$ and $q$ are coprime natural numbers.) [4 points]

When the random variable $X$ follows the binomial distribution $\mathrm { B } \left( n , \frac { 1 } { 2 } \right)$ and satisfies $\mathrm { E } \left( X ^ { 2 } \right) = \mathrm { V } ( X ) + 25$, what is the value of $n$? [3 points]
(1) 10
(2) 12
(3) 14
(4) 16
(5) 18

The random variable $X$ follows the binomial distribution $\mathrm { B } ( 80 , p )$ and $\mathrm { E } ( X ) = 20$. Find the value of $\mathrm { V } ( X )$. [3 points]

The random variable $X$ follows a binomial distribution $\mathrm { B } ( 80 , p )$ and $\mathrm { E } ( X ) = 20$. Find the value of $\mathrm { V } ( X )$. [3 points]

A die is rolled 5 times, and let $a$ be the number of times an odd number appears. A coin is tossed 4 times, and let $b$ be the number of times heads appears. If the probability that $a - b = 3$ is $\frac { q } { p }$, find the value of $p + q$. (Here, $p$ and $q$ are coprime natural numbers.) [3 points]

There is a bag containing 5 balls with the numbers $3,3,4,4,4$ written on them, one each. Using this bag and one die, a trial is performed to obtain a score according to the following rule.
A ball is randomly drawn from the bag. If the number on the drawn ball is 3, the die is rolled 3 times and the sum of the three numbers shown is the score. If the number on the drawn ball is 4, the die is rolled 4 times and the sum of the four numbers shown is the score.
What is the probability that the score obtained from one trial is 10 points? [4 points]
(1) $\frac { 13 } { 180 }$
(2) $\frac { 41 } { 540 }$
(3) $\frac { 43 } { 540 }$
(4) $\frac { 1 } { 12 }$
(5) $\frac { 47 } { 540 }$

A random variable $X$ follows a binomial distribution $\mathrm { B } \left( n , \frac { 1 } { 3 } \right)$ and $\mathrm { V} ( 2 X ) = 40$. What is the value of $n$? [3 points]
(1) 30
(2) 35
(3) 40
(4) 45
(5) 50

13. The defect rate of a batch of products is 0.02. Drawing one item at a time with replacement from this batch, 100 times total, let $X$ denote the number of defective items drawn. Then $D X = $ ______

Each member of a certain group uses mobile payment with probability $p$. The payment methods of each member are independent. Let $X$ be the number of people among 10 members of the group who use mobile payment. If $D(X) = 2.4$ and $P ( X = 4 ) < P ( X = 6 )$, then $p =$
A. 0.7
B. 0.6
C. 0.4
D. 0.3