This exercise is a multiple choice questionnaire. For each of the following questions, only one of the four proposed answers is correct.
A correct answer earns one point. An incorrect answer, a multiple answer, or the absence of an answer to a question earns or deducts no points.
PART I
In a mail processing centre, a machine is equipped with an automatic optical reader for recognizing postal addresses. This reading system correctly recognizes $97\%$ of addresses; the remaining mail, which will be described as unreadable for the machine, is directed to a centre employee responsible for reading the addresses. This machine has just read nine addresses. We denote by $X$ the random variable that gives the number of unreadable addresses among these nine addresses. We assume that $X$ follows the binomial distribution with parameters $n = 9$ and $p = 0.03$.

1. The probability that none of the nine addresses is unreadable is equal, to the nearest hundredth, to: a. 0 b. 1 c. 0.24 d. 0.76
2. The probability that exactly two of the nine addresses are unreadable for the machine is: a. $\binom{9}{2} \times 0.97^{2} \times 0.03^{7}$ b. $\binom{7}{2} \times 0.97^{2} \times 0.03^{7}$ c. $\binom{9}{2} \times 0.97^{7} \times 0.03^{2}$ d. $\binom{7}{2} \times 0.97^{7} \times 0.03^{2}$
3. The probability that at least one of the nine addresses is unreadable for the machine is: a. $P(X < 1)$ b. $P(X \leqslant 1)$ c. $P(X \geqslant 2)$ d. $1 - P(X = 0)$

PART II
An urn contains 5 green balls and 3 white balls, indistinguishable to the touch. We draw at random successively and without replacement two balls from the urn. We consider the following events:

• $V_{1}$: "the first ball drawn is green";
• $B_{1}$: "the first ball drawn is white";
• $V_{2}$: "the second ball drawn is green";
• $B_{2}$: "the second ball drawn is white".

1. The probability of $V_{2}$ given that $V_{1}$ is realized, denoted $P_{V_{1}}\left(V_{2}\right)$, is equal to: a. $\frac{5}{8}$ b. $\frac{4}{7}$ c. $\frac{5}{14}$ d. $\frac{20}{56}$
2. The probability of event $V_{2}$ is equal to: a. $\frac{5}{8}$ b. $\frac{5}{7}$ c. $\frac{3}{28}$ d. $\frac{9}{7}$

In basketball, it is possible to score baskets worth one point, two points or three points.
The coach of a basketball team decides to study the success statistics of his players' shots. He observes that during training, when Victor attempts a three-point shot, he succeeds with a probability of 0.32. During a training session, Victor makes a series of 15 three-point shots. We assume that these shots are independent.
Let $N$ be the random variable giving the number of baskets scored. The results of the requested probabilities should be, if necessary, rounded to the nearest thousandth.

1. We admit that the random variable $N$ follows a binomial distribution. Specify its parameters.
2. Calculate the probability that Victor succeeds in exactly 4 baskets during this series.
3. Determine the probability that Victor succeeds in at most 6 baskets during this series.
4. Determine the expected value of the random variable $N$.
5. Let $T$ be the random variable giving the number of points scored after this series of shots. a. Express $T$ as a function of $N$. b. Deduce the expected value of the random variable $T$. Give an interpretation of this value in the context of the exercise. c. Calculate $P ( 12 \leqslant T \leqslant 18 )$.

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) $\frac { 1 } { 2 }$

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 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) $\frac { 29 } { 72 }$

There are 30 True or False questions in an examination. A student knows the answer to 20 questions and guesses the answers to the remaining 10 questions at random. What is the probability that the student gets exactly 24 answers correct?
(A) $\frac{105}{2^9}$
(B) $\frac{105}{2^8}$
(C) $\frac{105}{2^{10}}$
(D) $\frac{4}{2^{10}}$

A pair of fair dice is thrown independently three times. The probability of getting a score of exactly 9 twice is
(1) $1 / 729$
(2) $8 / 9$
(3) $8 / 729$
(4) $8 / 243$

Each of the persons $A$ and $B$ independently tosses three fair coins. The probability that both of them get the same number of heads is: (1) $\frac { 5 } { 8 }$ (2) $\frac { 1 } { 8 }$ (3) $\frac { 5 } { 16 }$ (4) 1

Let P be a point in a plane with a coordinate system that is initially located at the origin $(0,0)$ and moves in the plane according to the following rule:
One dice is thrown. When the number on the dice is a multiple of three, point P moves 1 unit in the positive direction of the $x$-axis, and when the number on the dice is not a multiple of three, point P moves 1 unit in the positive direction of the $y$-axis.
Assume that the dice is thrown four times.
(1) The probability that P reaches point $(3,1)$ is $\frac{\mathbf{A}}{\mathbf{BC}}$.
(2) Altogether, the number of the points which P can reach is $\mathbf{D}$, and the coordinates of these points can be expressed in terms of an integer $k$ as
$$(k,\, \mathbf{E} - k) \quad (\mathbf{F} \leq k \leq \mathbf{G}).$$
Let us denote the probability that P can reach a given point $(k, \mathbf{E} - k)$ by $p_k$. Then the maximum value of $p_k$ is $\frac{\mathbf{HI}}{\mathbf{HI}}$, and the minimum value of $p_k$ is $\frac{\mathbf{J}}{\mathbf{BC}}$.
(3) The probability that $P$ passes through point $(1,1)$ and reaches point $(2,2)$ is $\frac{\mathbf{KL}}{\mathbf{BC}}$.

Consider a situation where products are produced sequentially. The events producing defective products are independent and identically distributed, and a defective product is produced with a probability of $\phi$ $(0 \leq \phi \leq 1)$. We consider the changes of the probability distributions before and after observing production results. In the following questions, $N (\geq 1)$ denotes the number of products observed.
By defining $v_i = 1$ if the $i$-th product is a defective product, and $v_i = 0$ if it is not defective, we get a series $\boldsymbol{v} = (v_1, \cdots, v_N)$, where the values can be 0 or 1. Let $N_d(\boldsymbol{v})$ be the number of observations with value of 1 in $\boldsymbol{v}$, obtain the occurrence probability of $\boldsymbol{v}$.