In a factory, two machines A and B are used to manufacture parts.
Machine A ensures $40\%$ of production and machine B ensures $60\%$. It is estimated that $10\%$ of parts from machine A have a defect and that $9\%$ of parts from machine B have a defect.
A part is chosen at random and we consider the following events:

• $A$: ``The part is produced by machine A''
• $B$: ``The part is produced by machine B''
• $D$: ``The part has a defect''
• $\bar{D}$: the opposite event of event $D$.

1. a. Translate the situation using a probability tree. b. Calculate the probability that the chosen part has a defect and was manufactured by machine A. c. Prove that the probability $P(D)$ of event $D$ is equal to 0.094. d. It is observed that the chosen part has a defect. What is the probability that this part comes from machine A?
2. It is estimated that machine A is properly adjusted if $90\%$ of the parts it manufactures are conforming. It is decided to check this machine by examining $n$ parts chosen at random ($n$ natural integer) from the production of machine A. These $n$ draws are treated as successive independent draws with replacement. We denote by $X_n$ the number of parts that are conforming in the sample of $n$ parts, and $F_n = \dfrac{X_n}{n}$ the corresponding proportion. a. Justify that the random variable $X_n$ follows a binomial distribution and specify its parameters. b. In this question, we take $n = 150$. Determine the asymptotic fluctuation interval $I$ at the $95\%$ threshold of the random variable $F_{150}$. c. A quality test counts 21 non-conforming parts in a sample of 150 parts produced. Does this call into question the adjustment of the machine? Justify the answer.

A garden centre sells young tree saplings that come from three horticulturists: $35 \%$ of the plants come from horticulturist $\mathrm { H } _ { 1 } , 25 \%$ from horticulturist $\mathrm { H } _ { 2 }$ and the rest from horticulturist $\mathrm { H } _ { 3 }$. Each horticulturist supplies two categories of trees: conifers and deciduous trees. The delivery from horticulturist $\mathrm { H } _ { 1 }$ contains $80 \%$ conifers while that from horticulturist $\mathrm { H } _ { 2 }$ contains only $50 \%$ and that from horticulturist $\mathrm { H } _ { 3 }$ only $30 \%$.
We consider the following events: $H _ { 1 }$ : ``the tree chosen was purchased from horticulturist $\mathrm { H } _ { 1 }$'', $H _ { 2 }$ : ``the tree chosen was purchased from horticulturist $\mathrm { H } _ { 2 }$'', $H _ { 3 }$ : ``the tree chosen was purchased from horticulturist $\mathrm { H } _ { 3 }$'', $C$ : ``the tree chosen is a conifer'', $F$ : ``the tree chosen is a deciduous tree''.

1. The garden centre manager chooses a tree at random from his stock.
   1. [a.] Construct a probability tree representing the situation.
   2. [b.] Calculate the probability that the tree chosen is a conifer purchased from horticulturist $\mathrm { H } _ { 3 }$.
   3. [c.] Justify that the probability of event $C$ is equal to 0.525.
   4. [d.] The tree chosen is a conifer. What is the probability that it was purchased from horticulturist $\mathrm { H } _ { 1 }$? Round to $10 ^ { - 3 }$.
2. A random sample of 10 trees is chosen from the stock of this garden centre. We assume that this stock is large enough that this choice can be treated as sampling with replacement of 10 trees from the stock. Let $X$ be the random variable that gives the number of conifers in the chosen sample.
   1. [a.] Justify that $X$ follows a binomial distribution and specify its parameters.
   2. [b.] What is the probability that the sample contains exactly 5 conifers? Round to $10 ^ { - 3 }$.
   3. [c.] What is the probability that this sample contains at least two deciduous trees? Round to $10 ^ { - 3 }$.

The company produces $40 \%$ of small-sized footballs and $60 \%$ of standard-sized footballs. It is admitted that $2 \%$ of small-sized footballs and $5 \%$ of standard-sized footballs do not comply with regulations. A football is chosen at random in the company.
Consider the events: $A$ : ``the football is small-sized'', $B$ : ``the football is standard-sized'', $C$ : ``the football complies with regulations'' and $\bar { C }$, the opposite event of C.

1. Represent this random experiment using a probability tree.
2. Calculate the probability that the football is small-sized and complies with regulations.
3. Show that the probability of event $C$ is equal to 0.962.
4. The football chosen does not comply with regulations. What is the probability that this football is small-sized? Round the result to $10 ^ { - 3 }$.

Machine A produces one third of the factory's sweets. The rest of production is ensured by machine B. When produced by machine B, the probability that a randomly selected sweet is deformed is equal to 0.02. In a quality control test, a sweet is randomly selected from the entire production. It is deformed.
What is the probability, rounded to the nearest hundredth, that it was produced by machine B?
Answer a: 0.02 Answer b: 0.67 Answer c: 0.44 Answer d: 0.01

A company packages white sugar from two farms $U$ and $V$ in 1 kg packets. We admit that $3\%$ of the sugar from farm $U$ is extra fine and that $5\%$ of the sugar from farm V is extra fine. A packet of sugar is randomly selected from the company's production and we consider the following events:

• $U$: ``The packet contains sugar from farm U'';
• $V$: ``The packet contains sugar from farm V'';
• $E$: ``The packet bears the label `extra fine' ''.

1. In this question, we admit that the company manufactures $30\%$ of its packets with sugar from farm U and the others with sugar from farm V, without mixing sugars from the two farms. a. What is the probability that the selected packet bears the label ``extra fine''? b. Given that a packet bears the label ``extra fine'', what is the probability that the sugar it contains comes from farm U?
2. The company wishes to modify its supply from the two farms so that among the packets bearing the label ``extra fine'', $30\%$ of them contain sugar from farm U. How should it supply itself from farms U and V? Any working will be valued in this question.

This club makes group orders of bearings for its members from two suppliers A and B.

• Supplier A offers higher prices but the bearings it sells are defect-free with a probability of 0.97.
• Supplier B offers more advantageous prices but its bearings are defective with a probability of 0.05.

A bearing is chosen at random from the club's stock and we consider the events: $A$: ``the bearing comes from supplier A'', $B$: ``the bearing comes from supplier B'', $D$: ``the bearing is defective''.

1. The club buys $40\%$ of its bearings from supplier A and the rest from supplier B. a. Calculate the probability that the bearing comes from supplier A and is defective. b. The bearing is defective. Calculate the probability that it comes from supplier B.
2. If the club wants less than $3.5\%$ of the bearings to be defective, what minimum proportion of bearings should it order from supplier A?

Exercise 1: Probability
A company manufactures components for the automotive industry. These components are designed on three assembly lines numbered 1 to 3.

• Half of the components are designed on line $\mathrm{n}^{\circ}1$;
• $30\%$ of the components are designed on line $\mathrm{n}^{\circ}2$;
• the remaining components are designed on line $\mathrm{n}^{\circ}3$.

At the end of the manufacturing process, it appears that $1\%$ of the parts from line $\mathrm{n}^{\circ}1$ have a defect, as do $0.5\%$ of the parts from line $\mathrm{n}^{\circ}2$ and $4\%$ of the parts from line $\mathrm{n}^{\circ}3$. One of these components is randomly selected. We denote:

• $C_1$ the event ``the component comes from line $\mathrm{n}^{\circ}1$'';
• $C_2$ the event ``the component comes from line $\mathrm{n}^{\circ}2$'';
• $C_3$ the event ``the component comes from line $\mathrm{n}^{\circ}3$'';
• $D$ the event ``the component is defective'' and $\bar{D}$ its complementary event.

Throughout the exercise, probability calculations will be given as exact decimal values or rounded to $10^{-4}$ if necessary.
PART A

1. Represent this situation with a probability tree.
2. Calculate the probability that the selected component comes from line $\mathrm{n}^{\circ}3$ and is defective.
3. Show that the probability of event $D$ is $P(D) = 0.0145$.
4. Calculate the probability that a defective component comes from line $\mathrm{n}^{\circ}3$.

PART B
The company decides to package the produced components by forming batches of $n$ units. We denote $X$ the random variable which, to each batch of $n$ units, associates the number of defective components in this batch. Given the company's production and packaging methods, we can consider that $X$ follows the binomial distribution with parameters $n$ and $p = 0.0145$.

1. In this question, the batches contain 20 units. We set $n = 20$. a. Calculate the probability that a batch contains exactly three defective components. b. Calculate the probability that a batch contains no defective components. Deduce the probability that a batch contains at least one defective component.
2. The company director wishes the probability of having no defective components in a batch of $n$ components to be greater than 0.85. He proposes to form batches of at most 11 components. Is he correct? Justify your answer.

PART C
The manufacturing costs of the components of this company are 15 euros if they come from assembly line $\mathrm{n}^{\circ}1$, 12 euros if they come from assembly line $\mathrm{n}^{\circ}2$ and 9 euros if they come from assembly line $\mathrm{n}^{\circ}3$. Calculate the average manufacturing cost of a component for this company.

Customs authorities are interested in imports of headphones bearing the logo of a certain brand. Customs seizures allow them to estimate that:

• $20 \%$ of headphones bearing this brand's logo are counterfeits;
• $2 \%$ of non-counterfeit headphones have a design defect;
• $10 \%$ of counterfeit headphones have a design defect.

The fraud agency randomly orders a headphone displaying the brand's logo from an internet site. Consider the following events:

• C: ``the headphone is counterfeit'';
• $D$: ``the headphone has a design defect'';
• $\bar { C }$ and $\bar { D }$ denote respectively the complementary events of $C$ and $D$.

Throughout the exercise, probabilities will be rounded to $10 ^ { - 3 }$ if necessary.

Part 1

1. Calculate $P ( C \cap D )$. You may use a probability tree.
2. Prove that $P ( D ) = 0,036$.
3. The headphone has a defect. What is the probability that it is counterfeit?

Part 2

We order $n$ headphones bearing this brand's logo. We treat this experiment as a random draw with replacement. Let $X$ be the random variable giving the number of headphones with a design defect in this batch.

1. In this question, $n = 35$. a. Justify that $X$ follows a binomial distribution $\mathscr { B } ( n , p )$ where $n = 35$ and $p = 0,036$. b. Calculate the probability that among the ordered headphones, exactly one has a design defect. c. Calculate $P ( X \leqslant 1 )$.
2. In this question, $n$ is not fixed. What is the minimum number of headphones to order so that the probability that at least one headphone has a defect is greater than 0.99?

Alice has two urns A and B each containing four indistinguishable balls. Urn A contains two green balls and two red balls. Urn B contains three green balls and one red ball. Alice randomly chooses an urn and then a ball from that urn. She obtains a green ball. The probability that she chose urn B is:
A. $\frac{3}{8}$
B. $\frac{1}{2}$
C. $\frac{3}{5}$
D. $\frac{5}{8}$

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 machine is defective. The probability that it is under warranty is approximately equal, to $10^{-3}$ near, to: a. 0.01 b. 0.024 c. 0.082 d. 0.1

A video game rewards players who have won a challenge with a randomly drawn object. The drawn object can be ``common'' or ``rare''. Two types of objects, common or rare, are available: swords and shields.
The video game designers have planned that:

• the probability of drawing a rare object is $7\%$;
• if a rare object is drawn, the probability that it is a sword is $80\%$;
• if a common object is drawn, the probability that it is a sword is $40\%$.

Part A
A player has just won a challenge and draws an object at random. We denote:

• R the event ``the player draws a rare object'';
• $E$ the event ``the player draws a sword'';
• $\bar{R}$ and $\bar{E}$ the complementary events of events $R$ and $E$.

1. Draw a probability tree modelling the situation, then calculate $P(R \cap E)$.
2. Calculate the probability of drawing a sword.
3. The player has drawn a sword. Determine the probability that it is a rare object. Round the result to the nearest thousandth.

A marketing agency studied customer satisfaction regarding customer service when purchasing a television. These purchases were made either online, in an appliance store chain, or in a large supermarket. Online purchases represent $60 \%$ of sales, appliance store purchases $30 \%$ of sales, and large supermarket purchases $10 \%$ of sales. A survey shows that the proportion of customers satisfied with customer service is:

• $75 \%$ for online customers;
• $90 \%$ for appliance store customers;
• $80 \%$ for large supermarket customers.

A customer who purchased the television model in question is chosen at random. The following events are defined:

• I: ``the customer made their purchase online'';
• $M$: ``the customer made their purchase in an appliance store'';
• $G$: ``the customer made their purchase in a large supermarket'';
• S: ``the customer is satisfied with customer service''.

If $A$ is any event, we denote by $\bar { A }$ its complementary event and $P ( A )$ its probability.

1. Reproduce and complete the tree diagram opposite.
2. Calculate the probability that the customer made their purchase online and is satisfied with customer service.
3. Prove that $P ( S ) = 0.8$.
4. A customer is satisfied with customer service. What is the probability that they made their purchase online? Give the result rounded to $10 ^ { - 3 }$.
5. To conduct the study, the agency must contact 30 customers each day among the television buyers. We assume that the number of customers is large enough to treat the choice of 30 customers as sampling with replacement. Let $X$ be the random variable that, for each sample of 30 customers, associates the number of customers satisfied with customer service. a. Justify that $X$ follows a binomial distribution and specify its parameters. b. Determine the probability, rounded to $10 ^ { - 3 }$, that at least 25 customers are satisfied in a sample of 30 customers contacted on the same day.
6. By solving an inequality, determine the minimum sample size of customers to contact so that the probability that at least one of them is not satisfied is greater than $0.99$.
7. In the two questions a. and b. that follow, we are only interested in online purchases. When a television order is placed by a customer, the delivery time of the television is modeled by a random variable $T$ equal to the sum of two random variables $T _ { 1 }$ and $T _ { 2 }$.

The random variable $T _ { 1 }$ models the integer number of days for the television to be transported from a storage warehouse to a distribution platform. The random variable $T _ { 2 }$ models the integer number of days for the television to be transported from this platform to the customer's home.
We admit that the random variables $T _ { 1 }$ and $T _ { 2 }$ are independent, and we are given:

• The expectation $E \left( T _ { 1 } \right) = 4$ and the variance $V \left( T _ { 1 } \right) = 2$;
• The expectation $E \left( T _ { 2 } \right) = 3$ and the variance $V \left( T _ { 2 } \right) = 1$. a. Determine the expectation $E ( T )$ and the variance $V ( T )$ of the random variable $T$. b. A customer places a television order online. Justify that the probability that they receive their television between 5 and 9 days after their order is greater than or equal to $\frac { 2 } { 3 }$.

To access a company's private network from outside, employee connections are randomly routed through three different remote servers, denoted $\mathrm{A}, \mathrm{B}$ and C. These servers have different technical characteristics and connections are distributed as follows:

• $25\%$ of connections are routed through server A;
• $15\%$ of connections are routed through server B;
• the remaining connections are made through server C.

A connection is said to be stable if the user does not experience a disconnection after authentication to the servers. The IT maintenance team has statistically observed that, under normal server operation:

• $90\%$ of connections via server A are stable;
• $80\%$ of connections via server B are stable;
• $85\%$ of connections via server C are stable.

Part A
We are interested in the state of a connection made by an employee of the company. We consider the following events:

• A: ``The connection was made via server A'';
• B: ``The connection was made via server B'';
• C: ``The connection was made via server C'';
• S: ``The connection is stable''.

We denote by $\bar{S}$ the complementary event of event $S$.

1. Copy and complete the weighted tree below modelling the situation described in the problem.
2. Prove that the probability that the connection is stable and passes through server B is equal to 0.12.
3. Calculate the probability $P(C \cap \bar{S})$ and interpret the result in the context of the exercise.
4. Prove that the probability of event $S$ is $P(S) = 0.855$.
5. Now suppose that the connection is stable. Calculate the probability that the connection was made from server B. Give the answer rounded to the nearest thousandth.

Part B
According to Part A, the probability that a connection is unstable is equal to 0.145.

1. In order to detect server malfunctions, we study a sample of 50 connections to the network, these connections being chosen at random. We assume that the number of connections is large enough that this choice can be treated as sampling with replacement.
   Let $X$ denote the random variable equal to the number of unstable connections to the company's network, in this sample of 50 connections. a. We admit that the random variable $X$ follows a binomial distribution. Specify its parameters. b. Give the probability that at most eight connections are unstable. Give the answer rounded to the nearest thousandth.
2. In this question, we now form a sample of $n$ connections, still under the same conditions, where $n$ denotes a strictly positive natural number. We denote by $X_n$ the random variable equal to the number of unstable connections and we admit that $X_n$ follows a binomial distribution with parameters $n$ and 0.145. a. Give the expression as a function of $n$ of the probability $p_n$ that at least one connection in this sample is unstable. b. Determine, by justifying, the smallest value of the natural number $n$ such that the probability $p_n$ is greater than or equal to 0.99.
3. We are interested in the random variable $F_n$ equal to the frequency of unstable connections in a sample of $n$ connections, where $n$ denotes a strictly positive natural number. We thus have $F_n = \frac{X_n}{n}$, where $X_n$ is the random variable defined in question 2. a. Calculate the expectation $E(F_n)$. We admit that $V(F_n) = \frac{0.123975}{n}$. b. Verify that: $P\left(\left|F_n - 0.145\right| \geqslant 0.1\right) \leqslant \frac{12.5}{n}$ c. A company manager studies a sample of 1000 connections and observes that for this sample $F_{1000} = 0.3$. He suspects a server malfunction. Is he right?

We have a bag and two urns A and B.

• The bag contains 4 balls: 1 ball with the letter A and 3 balls with the letter B.
• Urn A contains 5 tickets: 3 tickets of 50 euros and 2 tickets of 10 euros.
• Urn B contains 4 tickets: 1 ticket of 50 euros and 3 tickets of 10 euros.

A player randomly draws a ball from the bag:

• if it is a ball with the letter A, he randomly draws a ticket from urn A.
• if it is a ball with the letter B, he randomly draws a ticket from urn B.

We note the following events:

• $A$: the player obtains a ball with the letter A.
• $C$: the player obtains a 50 euro ticket.

1. Copy and complete the tree opposite representing the situation.
2. What is the probability of the event ``the player obtains a ball with the letter A and a ticket of $50 €$''?
3. Prove that the probability $P(C)$ is equal to 0.3375.
4. The player obtained a 10 euro ticket. Is the statement ``There is more than $80\%$ chance that he previously obtained a ball with the letter B'' true? Justify.
5. We denote $X_1$ the random variable that gives the sum, in euros, obtained by the player. Example: if the player obtains a 50 euro ticket, then $X_1 = 50$. Show that the expectation $E(X_1)$ is equal to 23.50 and that the variance $V(X_1)$ is equal to 357.75.
6. After returning the ball to the bag and the ticket to the urn from which it was taken, the player plays a second game. We denote $X_2$ the random variable that gives the sum obtained by the player in this second game. We denote $Y$ the random variable defined as follows: $Y = X_1 + X_2$. a. Show that $E(Y) = 47$. b. Explain why we have $V(Y) = V(X_1) + V(X_2)$.
7. The player plays likewise a third, fourth, \ldots, hundredth game. We thus define in the same way the random variables $X_3, X_4, \ldots, X_{100}$. We denote $Z$ the random variable defined by $Z = X_1 + X_2 + \ldots + X_{100}$. Prove that the probability that $Z$ belongs to the interval $]1950; 2750[$ is greater than or equal to 0.75.

10\% of the emails Cheol-su receives contain the word ``travel.'' 50\% of emails containing ``travel'' are advertisements, and 20\% of emails not containing ``travel'' are advertisements. Given that an email Cheol-su received is an advertisement, what is the probability that this email contains the word ``travel''? [3 points]
(1) $\frac { 5 } { 23 }$
(2) $\frac { 6 } { 23 }$
(3) $\frac { 7 } { 23 }$
(4) $\frac { 8 } { 23 }$
(5) $\frac { 9 } { 23 }$

10\% of the emails Chulsu received contain the word ``travel''. 50\% of emails containing ``travel'' are advertisements, and 20\% of emails not containing ``travel'' are advertisements. Given that an email Chulsu received is an advertisement, what is the probability that it contains the word ``travel''? [3 points]
(1) $\frac { 5 } { 23 }$
(2) $\frac { 6 } { 23 }$
(3) $\frac { 7 } { 23 }$
(4) $\frac { 8 } { 23 }$
(5) $\frac { 9 } { 23 }$

[Probability and Statistics] A training facility has three courses A, B, and C to be experienced in order, with the entrance and exit being the same. There are 30 envelopes at course A, 60 envelopes at course B, and 90 envelopes at course C. Each envelope contains 1, 2, or 3 coupons. The following table shows the number of envelopes by the number of coupons for each course.

\multicolumn{1}{|c|}{Number of Coupons}	1	2	3	Total
A	20	10	0	30
B	30	20	10	60
C	40	30	20	90

After completing each course, a student randomly selects one envelope from that course and receives the coupons inside. A student who started first completed all three courses and received a total of 4 coupons. What is the probability that the student received 2 coupons at course B? [3 points]
(1) $\frac { 6 } { 23 }$
(2) $\frac { 8 } { 23 }$
(3) $\frac { 10 } { 23 }$
(4) $\frac { 12 } { 23 }$
(5) $\frac { 14 } { 23 }$

Bag A contains 5 cards with the numbers $1,2,3,4,5$ written on them, and Bag B contains 6 cards with the numbers $1,2,3,4,5,6$ written on them. A die is rolled once. If the result is a multiple of 3, a card is randomly drawn from Bag A; otherwise, a card is randomly drawn from Bag B. Given that the number on the card drawn from the bag is even, what is the probability that the card was drawn from Bag A? [3 points]
(1) $\frac { 1 } { 5 }$
(2) $\frac { 2 } { 9 }$
(3) $\frac { 1 } { 4 }$
(4) $\frac { 2 } { 7 }$
(5) $\frac { 1 } { 3 }$

At a certain school, $60 \%$ of all students commute by bus, and the remaining $40 \%$ walk to school. Of the students who commute by bus, $\frac { 1 } { 20 }$ were late, and of the students who walk, $\frac { 1 } { 15 }$ were late. When one student is randomly selected from all students at this school and is found to be late, what is the probability that this student commuted by bus? [3 points]
(1) $\frac { 3 } { 7 }$
(2) $\frac { 9 } { 20 }$
(3) $\frac { 9 } { 19 }$
(4) $\frac { 1 } { 2 }$
(5) $\frac { 9 } { 17 }$

A bag contains some candies, $\frac { 2 } { 5 }$ of them are made of white chocolate and the remaining $\frac { 3 } { 5 }$ are made of dark chocolate. Out of the white chocolate candies, $\frac { 1 } { 3 }$ are wrapped in red paper, the rest are wrapped in blue paper. Out of the dark chocolate candies, $\frac { 2 } { 3 }$ are wrapped in red paper, the rest are wrapped in blue paper. If a randomly selected candy from the bag is found to be wrapped in red paper, then what is the probability that it is made up of dark chocolate?
(A) $\frac { 2 } { 3 }$
(B) $\frac { 3 } { 4 }$
(C) $\frac { 3 } { 5 }$
(D) $\frac { 1 } { 4 }$

A brand called Jogger's Pride produces pairs of shoes in three different units that are named $U _ { 1 } , U _ { 2 }$ and $U _ { 3 }$. These units produce $10 \% , 30 \% , 60 \%$ of the total output of the brand with the chance that a pair of shoes being defective is $20 \% , 40 \% , 10 \%$ respectively. If a randomly selected pair of shoes from the combined output is found to be defective, then what is the chance that the pair was manufactured in the unit $U _ { 3 }$?
(A) $30 \%$
(B) $15 \%$
(C) $\frac { 3 } { 5 } \times 100 \%$
(D) Cannot be determined from the given data.

A box $B _ { 1 }$ contains 1 white ball, 3 red balls and 2 black balls. Another box $B _ { 2 }$ contains 2 white balls, 3 red balls and 4 black balls. A third box $B _ { 3 }$ contains 3 white balls, 4 red balls and 5 black balls.
If 2 balls are drawn (without replacement) from a randomly selected box and one of the balls is white and the other ball is red, the probability that these 2 balls are drawn from box $B _ { 2 }$ is
(A) $\frac { 116 } { 181 }$
(B) $\frac { 126 } { 181 }$
(C) $\frac { 65 } { 181 }$
(D) $\frac { 55 } { 181 }$

There are three bags $B _ { 1 } , B _ { 2 }$ and $B _ { 3 }$. The bag $B _ { 1 }$ contains 5 red and 5 green balls, $B _ { 2 }$ contains 3 red and 5 green balls, and $B _ { 3 }$ contains 5 red and 3 green balls. Bags $B _ { 1 } , B _ { 2 }$ and $B _ { 3 }$ have probabilities $\frac { 3 } { 10 } , \frac { 3 } { 10 }$ and $\frac { 4 } { 10 }$ respectively of being chosen. A bag is selected at random and a ball is chosen at random from the bag. Then which of the following options is/are correct?
(A) Probability that the chosen ball is green, given that the selected bag is $B _ { 3 }$, equals $\frac { 3 } { 8 }$
(B) Probability that the chosen ball is green equals $\frac { 39 } { 80 }$
(C) Probability that the selected bag is $B _ { 3 }$, given that the chosen ball is green, equals $\frac { 5 } { 13 }$
(D) Probability that the selected bag is $B _ { 3 }$ and the chosen ball is green equals $\frac { 3 } { 10 }$

A student appears for a quiz consisting of only true-false type questions and answers all the questions. The student knows the answers of some questions and guesses the answers for the remaining questions. Whenever the student knows the answer of a question, he gives the correct answer. Assume that the probability of the student giving the correct answer for a question, given that he has guessed it, is $\frac { 1 } { 2 }$. Also assume that the probability of the answer for a question being guessed, given that the student's answer is correct, is $\frac { 1 } { 6 }$. Then the probability that the student knows the answer of a randomly chosen question is
(A) $\frac { 1 } { 12 }$
(B) $\frac { 1 } { 7 }$
(C) $\frac { 5 } { 7 }$
(D) $\frac { 5 } { 12 }$

A factory has a total of three manufacturing units, $M _ { 1 } , M _ { 2 }$, and $M _ { 3 }$, which produce bulbs independent of each other. The units $M _ { 1 } , M _ { 2 }$, and $M _ { 3 }$ produce bulbs in the proportions of $2 : 2 : 1$, respectively. It is known that $20 \%$ of the bulbs produced in the factory are defective. It is also known that, of all the bulbs produced by $M _ { 1 } , 15 \%$ are defective. Suppose that, if a randomly chosen bulb produced in the factory is found to be defective, the probability that it was produced by $M _ { 2 }$ is $\frac { 2 } { 5 }$.
If a bulb is chosen randomly from the bulbs produced by $M _ { 3 }$, then the probability that it is defective is $\_\_\_\_$.