A screw factory has two machines, I and II, for the production of a certain type of screw.
In September, machine I produced $\frac{54}{100}$ of the total screws produced by the factory. Of the screws produced by this machine, $\frac{25}{1000}$ were defective. In turn, $\frac{38}{1000}$ of the screws produced in the same month by machine II were defective.
The combined performance of the two machines is classified according to the table, in which $P$ indicates the probability of a randomly chosen screw being defective.
$$\begin{aligned} 0 \leq P < \frac{2}{100} & \quad \text{Excellent} \\ \frac{2}{100} \leq P < \frac{4}{100} & \quad \text{Good} \\ \frac{4}{100} \leq P < \frac{6}{100} & \quad \text{Fair} \\ \frac{6}{100} \leq P < \frac{8}{100} & \quad \text{Poor} \\ \frac{8}{100} \leq P \leq 1 & \quad \text{Very Poor} \end{aligned}$$
The combined performance of these machines in September can be classified as
(A) excellent. (B) good. (C) fair. (D) poor. (E) very poor.

Pouches A and B each contain five marbles with the numbers $1,2,3,4,5$ written on them, one number per marble. Chulsu draws one marble from pouch A, and Younghee draws one marble from pouch B. They check the numbers on the two marbles and do not put them back. This trial is repeated. What is the probability that the numbers on the two marbles drawn the first time are different, and the numbers on the two marbles drawn the second time are the same? [4 points]
(1) $\frac { 3 } { 20 }$
(2) $\frac { 1 } { 5 }$
(3) $\frac { 1 } { 4 }$
(4) $\frac { 3 } { 10 }$
(5) $\frac { 7 } { 20 }$

Box A contains 3 red balls and 5 black balls, and box B is empty. When 2 balls are randomly drawn from box A, if a red ball appears, perform [Execution 1], and if no red ball appears, perform [Execution 2]. What is the probability that the number of red balls in box B is 1? [3 points] [Execution 1] Put the drawn balls into box B. [Execution 2] Put the drawn balls into box B, and then randomly draw 2 more balls from box A and put them into box B.
(1) $\frac { 1 } { 2 }$
(2) $\frac { 7 } { 12 }$
(3) $\frac { 2 } { 3 }$
(4) $\frac { 3 } { 4 }$
(5) $\frac { 5 } { 6 }$

Bag A contains 2 white balls and 3 black balls, and Bag B contains 1 white ball and 3 black balls. One ball is randomly drawn from Bag A. If it is white, 2 white balls are put into Bag B; if it is black, 2 black balls are put into Bag B. Then one ball is randomly drawn from Bag B. What is the probability that the drawn ball is white? [4 points]
(1) $\frac { 1 } { 6 }$
(2) $\frac { 1 } { 5 }$
(3) $\frac { 7 } { 30 }$
(4) $\frac { 4 } { 15 }$
(5) $\frac { 3 } { 10 }$

With the notation of question 28, justify that, for all $j \in \llbracket 1 , n - 1 \rrbracket$, $$\mathbb { P } \left( C _ { j + 1 } \in \operatorname { Vect } \left( C _ { 1 } , \ldots , C _ { j } \right) \right) = \sum _ { \left( v _ { 1 } , \ldots , v _ { j } \right) \in \mathcal { V } _ { n , 1 } ^ { j } } \mathbb { P } \left( C _ { j + 1 } \in \operatorname { Vect } \left( v _ { 1 } , \ldots , v _ { j } \right) \right) \mathbb { P } \left( \left( C _ { 1 } = v _ { 1 } \right) \cap \cdots \cap \left( C _ { j } = v _ { j } \right) \right) .$$

One of the two boxes, box I and box II, was selected at random and a ball was drawn randomly out of this box. The ball was found to be red. If the probability that this red ball was drawn from box II is $\frac { 1 } { 3 }$, then the correct option(s) with the possible values of $n _ { 1 } , n _ { 2 } , n _ { 3 }$ and $n _ { 4 }$ is(are)
(A) $n _ { 1 } = 3 , n _ { 2 } = 3 , n _ { 3 } = 5 , n _ { 4 } = 15$
(B) $n _ { 1 } = 3 , n _ { 2 } = 6 , n _ { 3 } = 10 , n _ { 4 } = 50$
(C) $n _ { 1 } = 8 , n _ { 2 } = 6 , n _ { 3 } = 5 , n _ { 4 } = 20$
(D) $n _ { 1 } = 6 , n _ { 2 } = 12 , n _ { 3 } = 5 , n _ { 4 } = 20$

A ball is drawn at random from box I and transferred to box II. If the probability of drawing a red ball from box I, after this transfer, is $\frac { 1 } { 3 }$, then the correct option(s) with the possible values of $n _ { 1 }$ and $n _ { 2 }$ is(are)
(A) $\quad n _ { 1 } = 4$ and $n _ { 2 } = 6$
(B) $\quad n _ { 1 } = 2$ and $n _ { 2 } = 3$
(C) $n _ { 1 } = 10$ and $n _ { 2 } = 20$
(D) $n _ { 1 } = 3$ and $n _ { 2 } = 6$

A computer producing factory has only two plants $T_1$ and $T_2$. Plant $T_1$ produces $20\%$ and plant $T_2$ produces $80\%$ of the total computers produced. $7\%$ of computers produced in the factory turn out to be defective. It is known that $P$(computer turns out to be defective given that it is produced in plant $T_1$) $= 10P$(computer turns out to be defective given that it is produced in plant $T_2$), where $P(E)$ denotes the probability of an event $E$. A computer produced in the factory is randomly selected and it does not turn out to be defective. Then the probability that it is produced in plant $T_2$ is
(A) $\frac{36}{73}$
(B) $\frac{47}{79}$
(C) $\frac{78}{93}$
(D) $\frac{75}{83}$

A bag contains 4 red and 6 black balls. A ball is drawn at random from the bag, its color is observed and this ball along with two additional balls of the same color are returned to the bag. If now a ball is drawn at random from the bag, then the probability that this drawn ball is red, is:
(1) $\frac { 3 } { 4 }$
(2) $\frac { 3 } { 10 }$
(3) $\frac { 2 } { 5 }$
(4) $\frac { 1 } { 5 }$

An urn contains 5 red and 2 green balls. A ball is drawn at random from the urn. If the drawn ball is green, then a red ball is added to the urn and if the drawn ball is red, then a green ball is added to the urn; the original ball is not returned to the urn. Now, a second ball is drawn at random from it. The probability that the second ball is red, is:
(1) $\frac{21}{49}$
(2) $\frac{26}{49}$
(3) $\frac{32}{49}$
(4) $\frac{27}{49}$

An unbiased coin is tossed. If the outcome is a head then a pair of unbiased dice is rolled and the sum of the numbers obtained on them is noted. If the toss of the coin results in tail then a card from a well-shuffled pack of nine cards numbered $1,2,3 , \ldots , 9$ is randomly picked and the number on the card is noted. The probability that the noted number is either 7 or 8 is
(1) $\frac { 13 } { 36 }$
(2) $\frac { 19 } { 72 }$
(3) $\frac { 15 } { 72 }$
(4) $\frac { 19 } { 36 }$

A pack of cards has one card missing. Two cards are drawn randomly and are found to be spades. The probability that the missing card is not a spade, is :
(1) $\frac { 3 } { 4 }$
(2) $\frac { 52 } { 867 }$
(3) $\frac { 39 } { 50 }$
(4) $\frac { 22 } { 425 }$

An electric instrument consists of two units. Each unit must function independently for the instrument to operate. The probability that the first unit functions is 0.9 and that of the second unit is 0.8 . The instrument is switched on and it fails to operate. If the probability that only the first unit failed and second unit is functioning is $p$, then $98p$ is equal to

An urn contains 6 white and 9 black balls. Two successive draws of 4 balls are made without replacement. The probability, that the first draw gives all white balls and the second draw gives all black balls, is :
(1) $\frac { 5 } { 256 }$
(2) $\frac { 5 } { 715 }$
(3) $\frac { 3 } { 715 }$
(4) $\frac { 3 } { 256 }$

Bag 1 contains 4 white balls and 5 black balls, and Bag 2 contains $n$ white balls and 3 black balls. One ball is drawn randomly from Bag 1 and transferred to Bag 2. A ball is then drawn randomly from Bag 2. If the probability, that the ball drawn is white, is $29 / 45$, then $n$ is equal to :
(1) 6
(2) 3
(3) 5
(4) 4

A fair cubic die is rolled and it is known that one of its faces is in contact with the ground.
Given this, what is the probability that only one of the corners A and B is in contact with the ground?
A) $\frac { 1 } { 2 }$
B) $\frac { 1 } { 3 }$
C) $\frac { 2 } { 3 }$
D) $\frac { 1 } { 6 }$
E) $\frac { 5 } { 6 }$

Two different digits are randomly selected from the set $A = \{ 1,2,3,4,5,6,7 \}$.
Given that the product of the selected digits is an even number, what is the probability that the sum of these digits is also an even number?
A) $\frac { 1 } { 2 }$
B) $\frac { 1 } { 3 }$
C) $\frac { 1 } { 4 }$
D) $\frac { 1 } { 5 }$
E) $\frac { 1 } { 6 }$

A certain bus arrives at the bus stop near Duru's house with probability $\dfrac{7}{10}$ at exactly 09:02 and with probability $\dfrac{3}{10}$ at exactly 09:03. Duru leaves home at exactly 09:00 to catch this bus. The time it takes for Duru to reach the stop is 100 seconds with probability $\dfrac{1}{2}$, 150 seconds with probability $\dfrac{3}{10}$, and 250 seconds with probability $\dfrac{1}{5}$.
What is the probability that Duru is at the stop when the bus arrives?
A) $\dfrac{55}{100}$ B) $\dfrac{59}{100}$ C) $\dfrac{63}{100}$ D) $\dfrac{67}{100}$ E) $\dfrac{71}{100}$

In front of the two doors of a shopping mall, there are 2 parking lots named Blue and Red in front of the first door, and 3 parking lots named Yellow, Orange and Green in front of the second door. Kartal, who came to this shopping mall, randomly came in front of one of the doors and randomly parked his car in one of the parking lots in front of that door and entered the shopping mall. When leaving the shopping mall, since Kartal forgot which parking lot he parked his car in and which door he entered the shopping mall from, he exited from one of the doors randomly and searched for his car in one of the parking lots in front of that door randomly.
Accordingly, what is the probability that the parking lot where Kartal searched for his car is the parking lot where he parked his car?
A) $\frac{1}{5}$ B) $\frac{5}{24}$ C) $\frac{6}{25}$ D) $\frac{7}{36}$ E) $\frac{11}{48}$