A company manufactures microchips. Each chip can have two defects denoted A and B.
A statistical study shows that $2.8\%$ of chips have defect A, $2.2\%$ of chips have defect B, and fortunately, $95.4\%$ of chips have neither of the two defects.
The probability that a randomly selected chip has both defects is: a. 0.05 b. 0.004 c. 0.046 d. We cannot know

Exercise 4 — 7 points
Theme: Probability
During the manufacture of a pair of glasses, the pair of lenses must undergo two treatments denoted T1 and T2.
Part A
A pair of lenses is randomly selected from production. We denote by $A$ the event: ``the pair of lenses has a defect for treatment T1''. We denote by $B$ the event: ``the pair of lenses has a defect for treatment T2''. We denote by $\bar{A}$ and $\bar{B}$ respectively the complementary events of $A$ and $B$.
A study has shown that:

• the probability that a pair of lenses has a defect for treatment T1, denoted $P(A)$, is equal to 0.1.
• the probability that a pair of lenses has a defect for treatment T2, denoted $P(B)$, is equal to 0.2.
• the probability that a pair of lenses has neither of the two defects is 0.75.

1. Copy and complete the following table with the corresponding probabilities.
   

   	$A$	$\bar{A}$	Total
   $B$
   $\bar{B}$
   Total			1

   
   
2. a. Determine, by justifying the answer, the probability that a pair of lenses, randomly selected from production, has a defect for at least one of the two treatments T1 or T2. b. Give the probability that a pair of lenses, randomly selected from production, has two defects, one for each treatment T1 and T2. c. Are the events $A$ and $B$ independent? Justify the answer.
   
3. Calculate the probability that a pair of lenses, randomly selected from production, has a defect for only one of the two treatments.
   
4. Calculate the probability that a pair of lenses, randomly selected from production, has a defect for treatment T2, given that this pair of lenses has a defect for treatment T1.

Part B
A sample of 50 pairs of lenses is randomly selected from production. We assume that the production is large enough to assimilate this selection to a draw with replacement. We denote by $X$ the random variable which, to each sample of this type, associates the number of pairs of lenses that have the defect for treatment T1.

1. Justify that the random variable $X$ follows a binomial distribution and specify the parameters of this distribution.
   
2. Give the expression allowing the calculation of the probability of having, in such a sample, exactly 10 pairs of lenses that have this defect. Perform this calculation and round the result to $10^{-3}$.
   
3. On average, how many pairs of lenses with this defect can be found in a sample of 50 pairs?

A technician controls the machines equipping a large company. All these machines are identical. We know that:

• $20\%$ of machines are under warranty;
• $0.2\%$ of machines are both defective and under warranty;
• $8.2\%$ of machines are defective.

The technician tests a machine at random. We consider the following events:

• G: ``the machine is under warranty'';
• $D$: ``the machine is defective'';
• $\bar{G}$ and $\bar{D}$ denote respectively the complementary events of $G$ and $D$.

The probability $p(\bar{G} \cap D)$ is equal to: a. 0.01 b. 0.08 c. 0.1 d. 0.21

Question 171
Uma pesquisa mostrou que, em um grupo de 200 pessoas, 120 gostam de futebol, 90 gostam de vôlei e 40 gostam de ambos os esportes. O número de pessoas que não gostam de nenhum dos dois esportes é
(A) 10 (B) 20 (C) 30 (D) 40 (E) 50

Uma pesquisa realizada com 200 pessoas investigou a preferência por três tipos de esporte: futebol, vôlei e basquete. Os resultados mostraram que 120 pessoas preferem futebol, 80 preferem vôlei e 60 preferem basquete. Algumas pessoas indicaram mais de uma preferência: 30 preferem futebol e vôlei, 20 preferem futebol e basquete, 10 preferem vôlei e basquete, e 5 preferem os três esportes. Quantas pessoas não preferem nenhum dos três esportes?
(A) 5 (B) 10 (C) 15 (D) 20 (E) 25

Em uma sala de aula com 30 alunos, 18 estudam Matemática, 15 estudam Física e 8 estudam ambas as disciplinas. O número de alunos que não estudam nenhuma das duas disciplinas é
(A) 3 (B) 5 (C) 7 (D) 10 (E) 12

In a class of 30 students, 18 study mathematics, 15 study physics, and 8 study both subjects. How many students study neither mathematics nor physics?
(A) 3
(B) 4
(C) 5
(D) 6
(E) 7

Each student in a small school has to be a member of at least one of THREE school clubs. It is known that each club has 35 members. It is not known how many students are members of two of the three clubs, but it is known that exactly 10 students are members of all three clubs. What is the largest possible total number of students in the school? What is the smallest possible total number of students in the school?

The universal set is $U = \{ x \mid x$ is a natural number not exceeding 9 $\}$, and two subsets of $U$ are $$A = \{ 3,6,7 \} , B = \{ a - 4,8,9 \}$$ If $$A \cap B ^ { C } = \{ 6,7 \}$$ find the value of the natural number $a$. [3 points]

For the universal set $U = \{ 1,2,3,4,5,6,7,8 \}$ and two subsets $$A = \{ 1,2,3 \} , \quad B = \{ 2,4,6,8 \}$$ Find the value of $n \left( A \cup B ^ { C } \right)$. [3 points]

For two events $A$ and $B$, $A$ and $B ^ { C }$ are mutually exclusive events, and $$\mathrm { P } ( A ) = \frac { 1 } { 3 } , \mathrm { P } \left( A ^ { C } \cap B \right) = \frac { 1 } { 6 }$$ What is the value of $\mathrm { P } ( B )$? (Here, $A ^ { C }$ is the complement of $A$.) [3 points]
(1) $\frac { 5 } { 12 }$
(2) $\frac { 1 } { 2 }$
(3) $\frac { 7 } { 12 }$
(4) $\frac { 2 } { 3 }$
(5) $\frac { 3 } { 4 }$

For two events $A , B$, $$\mathrm { P } \left( A ^ { C } \right) = \frac { 2 } { 3 } , \quad \mathrm { P } \left( A ^ { C } \cap B \right) = \frac { 1 } { 4 }$$ What is the value of $\mathrm { P } ( A \cup B )$? (Note: $A ^ { C }$ is the complement of $A$.) [3 points]
(1) $\frac { 1 } { 2 }$
(2) $\frac { 7 } { 12 }$
(3) $\frac { 2 } { 3 }$
(4) $\frac { 3 } { 4 }$
(5) $\frac { 5 } { 6 }$

A bag contains 10 cards with natural numbers from 1 to 10 written on them, one number per card. When drawing 3 cards simultaneously at random from the bag, what is the probability that the smallest of the three natural numbers on the drawn cards is at most 4 or at least 7? [3 points]
(1) $\frac { 4 } { 5 }$
(2) $\frac { 5 } { 6 }$
(3) $\frac { 13 } { 15 }$
(4) $\frac { 9 } { 10 }$
(5) $\frac { 14 } { 15 }$

A bag contains 5 white balls with the numbers $1,2,3,4,5$ written on them one each, and 5 black balls with the numbers $2,3,4,5,6$ written on them one each. When 2 balls are drawn simultaneously at random from the bag, what is the probability that the 2 balls drawn are either the same color or have the same number written on them? [3 points]
(1) $\frac { 7 } { 15 }$
(2) $\frac { 8 } { 15 }$
(3) $\frac { 3 } { 5 }$
(4) $\frac { 2 } { 3 }$
(5) $\frac { 11 } { 15 }$

If set $A = \{ x \mid -5 < x < 2 \}$, $B = \{ x \mid -3 < x < 3 \}$, then $A \cap B =$

2. Let the universal set $U = \{ 1,2,3,4,5,6 \} , A = \{ 1,2 \} , B = \{ 2,3,4 \}$. Then $A \cap \left( C _ { U } B \right) =$
(A) $\{ 1,2,5,6 \}$
(B) $\{ 1 \}$
(C) $\{ 2 \}$
(D) $\{ 1,2,3,4 \}$

Journey to the West, Romance of the Three Kingdoms, Water Margin, and Dream of the Red Chamber are treasures of classical Chinese literature, collectively known as the Four Great Classical Novels of China. To understand the reading situation of these four classics among students in a school, a random survey was conducted of 100 students. Among them, 90 students had read either Journey to the West or Dream of the Red Chamber, 80 students had read Dream of the Red Chamber, and 60 students had read both Journey to the West and Dream of the Red Chamber. The estimated value of the ratio of the number of students who have read Journey to the West to the total number of students in the school is
A. 0.5
B. 0.6
C. 0.7
D. 0.8

Let $s > 1$ be a real number and let $X$ be a random variable taking values in $\mathbb{N}^*$ following the zeta distribution with parameter $s$.
Show that, for $r \in \mathbb{N}^*$, $k_1 < \cdots < k_r$ in $\mathbb{N}^*$ and $\left(n_1, \ldots, n_r\right) \in \mathbb{N}^r$, we have $$\begin{aligned} & P\left(\nu_{p_{k_1}}(X) = n_1, \ldots, \nu_{p_{k_r}}(X) = n_r\right) = \\ & \sum_{\ell=0}^{r}(-1)^{\ell} \sum_{\substack{\left(\varepsilon_1, \ldots, \varepsilon_r\right) \in \{0,1\}^r \\ \varepsilon_1 + \cdots + \varepsilon_r = \ell}} P\left(\nu_{p_{k_1}}(X) \geqslant n_1 + \varepsilon_1, \nu_{p_{k_2}}(X) \geqslant n_2 + \varepsilon_2, \ldots, \nu_{p_{k_r}}(X) \geqslant n_r + \varepsilon_r\right). \end{aligned}$$

Exercise VI

$\Omega$ denotes the sample space of a random experiment E and P denotes a probability on $\Omega$. $A$ and $B$ are two events with probabilities $0.6$ and $0.4$ respectively. We further assume that $P ( A \cup B ) = 0.8$. VI-A- $\quad P ( A \cap B ) = 0.24$. VI-B- $\quad A$ and $B$ are complementary events. VI-C- $\quad A$ and $B$ are independent events. VI-D- $\quad A$ and $B$ are mutually exclusive events.
For each statement, indicate whether it is TRUE or FALSE.

Find the number of positive integers less than or equal to 6300 which are not divisible by 3, 5 and 7.

In a class of 45 students, three students can write well using either hand. The number of students who can write well only with the right hand is 24 more than the number of those who write well only with the left hand. Then, the number of students who can write well with the right hand is:
(A) 33
(B) 36
(C) 39
(D) 41

In a study about a pandemic, data of 900 persons was collected. It was found that
190 persons had symptom of fever, 220 persons had symptom of cough, 220 persons had symptom of breathing problem, 330 persons had symptom of fever or cough or both, 350 persons had symptom of cough or breathing problem or both, 340 persons had symptom of fever or breathing problem or both, 30 persons had all three symptoms (fever, cough and breathing problem). If a person is chosen randomly from these 900 persons, then the probability that the person has at most one symptom is $\_\_\_\_$.

Let $X$ and $Y$ are two events such that $P ( X \cup Y ) = P ( X \cap Y )$. Statement 1: $P ( X \cap Y' ) = P ( X' \cap Y ) = 0$. Statement 2: $P ( X ) + P ( Y ) = 2 P ( X \cap Y )$
(1) Statement 1 is false, Statement 2 is true.
(2) Statement 1 is true, Statement 2 is true, Statement 2 is not a correct explanation of Statement 1.
(3) Statement 1 is true, Statement 2 is false.
(4) Statement 1 is true, Statement 2 is true; Statement 2 is a correct explanation of Statement 1.

In a class of 140 students numbered 1 to 140, all even numbered students opted Mathematics course, those whose number is divisible by 3 opted Physics course and those whose number is divisible by 5 opted Chemistry course. Then the number of students who did not opt for any of the three courses is:
(1) 42
(2) 1
(3) 38
(4) 102

Let $\cup _ { i = 1 } ^ { 50 } X _ { i } = \cup _ { i = 1 } ^ { n } Y _ { i } = T$, where each $X _ { i }$ contains 10 elements and each $Y _ { i }$ contains 5 elements. If each element of the set $T$ is an element of exactly 20 of sets $X _ { i }$'s and exactly 6 of sets $Y _ { i }$'s then $n$ is equal to:
(1) 15
(2) 50
(3) 45
(4) 30