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

A couple, both 30 years old, intends to take out a private pension plan. The insurance company researched, to define the value of the monthly contribution, estimates the probability that at least one of them will be alive in 50 years, based on population data, which indicate that 20\% of men and 30\% of women today will reach the age of 80.
What is this probability?
(A) $50\%$
(B) $44\%$
(C) $38\%$
(D) $25\%$
(E) $6\%$

For two events $A$ and $B$ in the sample space $S$, if they are mutually exclusive events, $A \cup B = S$, and $\mathrm { P } ( A ) = 2 \mathrm { P } ( B )$, what is the value of $\mathrm { P } ( A )$? [3 points]
(1) $\frac { 2 } { 3 }$
(2) $\frac { 1 } { 2 }$
(3) $\frac { 2 } { 5 }$
(4) $\frac { 1 } { 3 }$
(5) $\frac { 1 } { 4 }$

Two events $A$ and $B$ are mutually exclusive, and $$\mathrm { P } ( A ) = \mathrm { P } ( B ) , \quad \mathrm { P } ( A ) \mathrm { P } ( B ) = \frac { 1 } { 9 }$$ What is the value of $\mathrm { P } ( A \cup B )$? [3 points]
(1) $\frac { 1 } { 6 }$
(2) $\frac { 1 } { 3 }$
(3) $\frac { 1 } { 2 }$
(4) $\frac { 2 } { 3 }$
(5) $\frac { 5 } { 6 }$

A student named Chulsu participated in a design competition. Participants receive scores in two categories, and the possible scores in each category are one of three types shown in the table. The probability that Chulsu receives score A in each category is $\frac { 1 } { 2 }$, the probability of receiving score B is $\frac { 1 } { 3 }$, and the probability of receiving score C is $\frac { 1 } { 6 }$. When the event of receiving audience voting scores and the event of receiving judge scores are mutually independent, what is the probability that the sum of the two scores Chulsu receives is 70? [3 points]

Category	Score A	Score B	Score C
Audience Voting	40	30	20
Judges	50	40	30

(1) $\frac { 1 } { 3 }$
(2) $\frac { 11 } { 36 }$
(3) $\frac { 5 } { 18 }$
(4) $\frac { 1 } { 4 }$
(5) $\frac { 2 } { 9 }$

Chulsu participated in a certain design competition. Participants receive scores in two categories, and the possible scores in each category are one of three types shown in the table. The probability that Chulsu receives score A in each category is $\frac { 1 } { 2 }$, the probability of receiving score B is $\frac { 1 } { 3 }$, and the probability of receiving score C is $\frac { 1 } { 6 }$. When the event of receiving an audience vote score and the event of receiving a judge score are independent, what is the probability that the sum of the two scores Chulsu receives is 70? [3 points]

Category	Score A	Score B	Score C
Audience Vote	40	30	20
Judge	50	40	30

(1) $\frac { 1 } { 3 }$
(2) $\frac { 11 } { 36 }$
(3) $\frac { 5 } { 18 }$
(4) $\frac { 1 } { 4 }$
(5) $\frac { 2 } { 9 }$

For two events $A , B$, $A ^ { C }$ and $B$ are mutually exclusive events, and $$\mathrm { P } ( A ) = 2 \mathrm { P } ( B ) = \frac { 3 } { 5 }$$ When this condition is satisfied, what is the value of $\mathrm { P } \left( A \cap B ^ { C } \right)$? (Here, $A ^ { C }$ is the complement of $A$.) [3 points]
(1) $\frac { 3 } { 20 }$
(2) $\frac { 1 } { 5 }$
(3) $\frac { 1 } { 4 }$
(4) $\frac { 3 } { 10 }$
(5) $\frac { 7 } { 20 }$

For two events $A$ and $B$, $$\mathrm { P } ( A \cap B ) = \frac { 1 } { 8 } , \mathrm { P } \left( A \cap B ^ { C } \right) = \frac { 3 } { 16 }$$ What is the value of $\mathrm { P } ( A )$? (Here, $B ^ { C }$ is the complement of $B$.) [3 points]
(1) $\frac { 3 } { 16 }$
(2) $\frac { 7 } { 32 }$
(3) $\frac { 1 } { 4 }$
(4) $\frac { 9 } { 32 }$
(5) $\frac { 5 } { 16 }$

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 box contains 5 white masks and 9 black masks. When 3 masks are randomly drawn simultaneously from the box, what is the probability that at least one of the 3 masks is white? [3 points]
(1) $\frac { 8 } { 13 }$
(2) $\frac { 17 } { 26 }$
(3) $\frac { 9 } { 13 }$
(4) $\frac { 19 } { 26 }$
(5) $\frac { 10 } { 13 }$

9. A card is randomly drawn from a shuffled deck of playing cards (52 cards). Event $A$ is ``drawing the King of Hearts'', and event $B$ is ``drawing a Spade''. Then the probability $P ( A \cup B ) = $ $\_\_\_\_$ $\frac { 7 } { 26 }$ (express the result as a fraction in lowest terms).
Analysis: This examines the probability formula for mutually exclusive events. $P ( A \cup B ) = \frac { 1 } { 52 } + \frac { 13 } { 52 } = \frac { 7 } { 26 }$

In a municipality there are 6250 households, of which 2250 have a fast internet connection. Two thirds of the households that have a fast internet connection also have a subscription to a streaming service. $46 \%$ of all households have neither a fast internet connection nor a subscription to a streaming service.
Consider the following events:\ $A$ : ``A randomly selected household has a fast internet connection.''\ $B$ : ``A randomly selected household has a subscription to a streaming service.''\ Create a completely filled four-field table for the described situation and check whether the events $A$ and $B$ are stochastically independent.

Determine $x$ using a four-field table.\n(for verification: $x = 13$ )

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 $s$. If $n \in \mathbb{N}^*$, we denote $\{n \mid X\}$ the event ``$n$ divides $X$'' and $\{n \nmid X\}$ the complementary event.
Calculate $P(n \mid X)$ for $n \in \mathbb{N}^*$.

Let $G _ { 0 } = \left( S _ { 0 } , A _ { 0 } \right)$ be a particular fixed graph with $s _ { 0 } = s _ { G _ { 0 } }$, $a _ { 0 } = a _ { G _ { 0 } }$, $s_0 \geq 2$, $a_0 \geq 1$. For a graph $H = \left( S _ { H } , A _ { H } \right)$ with $S _ { H } \subset \llbracket 1 , n \rrbracket$, the random variable $X_H$ is defined by: $$\forall G \in \Omega _ { n } \quad X _ { H } ( G ) = \begin{cases} 1 & \text { if } H \subset G \\ 0 & \text { otherwise } \end{cases}$$ Show that $$\mathbf { E } \left( X _ { H } \right) = p _ { n } ^ { a _ { H } } .$$

We fix $n \in \mathbf { N } ^ { * }$ and draw successively and with replacement two integers $p$ and $q$ according to a uniform distribution on $\llbracket 1 , n \rrbracket$. The event $E_n$ is defined as "We obtain $(p,q) \in E_1 \cup E_2 \cup E_3$".
Using the result $\mathbf{P}(A_n \cup B_n) \sim \dfrac{\ln n}{n}$ as $n \to +\infty$, deduce $$\lim _ { n \rightarrow + \infty } \mathbf { P } \left( E _ { n } \right) .$$

148. If $A$ and $B$ are two events from sample space $S$ such that $P(A) = 0.6$, $P(B) = 0.7$, and $P(A \cap B') = 0.2$, then $P(A' \cap B)$ is equal to?
(1) $0.1$ (2) $0.3$ (3) $0.4$ (4) $0.5$

148- Three people are working on decoding a message. Their probabilities of success are $\frac{2}{3}$, $\frac{3}{4}$, and $\frac{1}{2}$ respectively. What is the probability that at least one of them succeeds?
(1) $\dfrac{19}{24}$ (2) $\dfrac{5}{6}$ (3) $\dfrac{11}{12}$ (4) $\dfrac{23}{24}$

146- We have three urns. The first urn contains 9 white and 4 black marbles, the second contains 9 black and 4 white marbles, and the third contains 5 white and 5 black marbles. One marble is randomly drawn from one urn. What is the probability that at least one of these two marbles is black?
\[ (1)\quad \frac{1}{3} \qquad (2)\quad \frac{11}{18} \qquad (3)\quad \frac{25}{36} \qquad (4)\quad \frac{13}{18} \]
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126-- In a random experiment, $S=\{x,y,z\}$ is a sample space. If $P(x)$, $P(y)$, and $P(z)$ form a geometric sequence and together they are less than one unit and their geometric mean is $\dfrac{1}{5}$, then the smallest simple event in $S$ is how much?
(1) $\dfrac{2-\sqrt{2}}{5}$ (2) $\dfrac{2-\sqrt{2}}{5}$ (3) $\dfrac{2-\sqrt{3}}{10}$ (4) $\dfrac{2-\sqrt{3}}{10}$

31. The probabilities that a student passes in Mathematics, Physics and Chemistry are $\mathrm { m } , \mathrm { p }$ and c , respectively. Of these subjects, the student has a $75 \%$ chance of passing in atleast one, a $50 \%$ chance of passing in atleast two, and a $40 \%$ chance of passing in exactly two. Which of the following relations are true?
(A) $\mathrm { p } + \mathrm { m } + \mathrm { c } = 19 / 20$
(B) $p + m + c = 27 / 20$
(C) $\mathrm { pmc } = 1 / 10$
(D) $\mathrm { pmc } = 1 / 4$

For a student to qualify, he must pass at least two out of three exams. The probability that he will pass the 1st exam is P . If he fails in one of the exams then the probability of his passing in the next exam is $\mathrm { P } / 2$ otherwise it remains the same. Find the probability that he will qualify.

Four persons independently solve a certain problem correctly with probabilities $\frac { 1 } { 2 } , \frac { 3 } { 4 } , \frac { 1 } { 4 } , \frac { 1 } { 8 }$. Then the probability that the problem is solved correctly by at least one of them is
(A) $\frac { 235 } { 256 }$
(B) $\frac { 21 } { 256 }$
(C) $\frac { 3 } { 256 }$
(D) $\frac { 253 } { 256 }$