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.

17. One day, Captain Haddock receives a mysterious letter with a confusing paragraph. Captain Haddock and Tintin are investigating the matter. There are two possible suspects: Professor Calculus and Thomson $\&$ Thompson. Based on their past experience:

• The probability that Professor Calculus sends a letter is $60 \%$, while the probability that Thomson $\&$ Thompson send a letter is $40 \%$.
• When Professor Calculus sends letters, there is an $80 \%$ probability that the letter contains a confusing paragraph.
• When Thomson \& Thompson send letters, there is a $5 \%$ probability that the letter contains a confusing paragraph.

What is the probability that the letter was sent by Professor Calculus?
(a) 0.96
(b) 0.80
(c) 0.50
(d) 0.48

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) $\frac { 3 } { 10 }$

In Sunnytown there are 6000 single-family houses, of which 2400 are equipped with a wood pellet heating system. Two thirds of the single-family houses with wood pellet heating have this combined with a solar thermal system. 50\% of all single-family houses are equipped with neither a wood pellet heating system nor a solar thermal system.
(1a) [3 marks] Create a completely filled four-field table for the described situation.
(1b) [2 marks] A randomly selected single-family house is equipped with a solar thermal system. What is the probability that it has a wood pellet heating system?
The tree diagram shown represents a two-stage random experiment with events $A$ and $B$ as well as their complementary events $\bar { A }$ and $\bar { B }$. [Figure]
(2a) [2 marks] Determine the value of $p$ so that event $B$ occurs in this random experiment with probability 0.3.
(2b) [3 marks] Determine the maximum possible value that the probability of $B$ can assume.
On a section of a lightly travelled country road, a maximum speed of 80 km/h is permitted. At one location on this section, the speed of passing cars is measured. In the following, only those journeys are considered where the drivers were able to choose their speed independently of one another.
For the first 200 recorded journeys, the following distribution was obtained after classification into speed classes: [Figure]
In 62\% of the 200 journeys, the driver was travelling alone, 65 of these solo drivers exceeded the speed limit. One journey is randomly selected from the 200 journeys. The following events are considered: $A$ : ``The driver was travelling alone.'' $S$ : ``The car was speeding.''

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

154. Five white marbles numbered 1 to 5 and five black marbles numbered 1 to 5 are placed in two separate containers. Two marbles are drawn at random from each container. If the sum of the two marbles from each container is 6, what is the probability that both marbles are the same color?
(1) $\dfrac{2}{5}$ (2) $\dfrac{4}{9}$ (3) $\dfrac{5}{9}$ (4) $\dfrac{3}{5}$

155. In two containers there are respectively 24 and 18 identical balls. In the first container there are 6 white balls and in the second container there are 3 white balls. From the first container 7 balls and from the second container 5 balls are randomly drawn and placed in another container. Then from the last container one ball is drawn. What is the probability that this ball is white?
(1) $\dfrac{13}{72}$ (2) $\dfrac{7}{36}$
(3) $\dfrac{15}{72}$ (4) $\dfrac{31}{144}$
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136. In the first jar there are 3 blue marbles and 6 red marbles, and in the second jar there are 4 blue marbles and 5 red marbles. Two coins are tossed. If the total number of heads is more than 9, one marble is taken from the first jar and added to the second jar. Otherwise, one marble is taken from the second jar and added to the first jar. Now one marble is selected from the jar with more marbles. What is the probability that this marble is red?
(1) $\dfrac{157}{270}$ (2) $\dfrac{165}{270}$ (3) $\dfrac{173}{270}$ (4) $\dfrac{185}{270}$

24 -- A device is designed so that it randomly receives one of two letters A or B as input and passes through three stages. At each stage, the input letter is printed with probability $\frac{1}{r}$ without change, or it moves to the next stage changed to the other letter. If the probability of selecting letter A as input is 2 times that of letter B, with what probability is ``A'' printed by the device equal to the probability of the input letter being A?
(1) $\dfrac{14}{23}$ (2) $\dfrac{21}{22}$ (3) $\dfrac{9}{41}$ (4) $\dfrac{17}{41}$
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10. If from each of the three boxes containing 3 white and 1 black, 2 white and 2 black, 1 white and 3 black balls, one ball is drawn at random, then the probability that 2 while and 1 black ball will be drawn is:
(A) $13 / 32$
(B) $1 / 4$
(C) $1 / 32$
(D) $3 / 16$

31. A fair coin is tossed repeatedly. If tail appears on first four tosses, then the probability of head appearing on fifth toss equals:
(A) $1 / 2$
(B) $1 / 32$
(C) $31 / 32$
(D) $1 / 5$

14. (a) An urn contains $m$ white and $n$ black balls. A ball is drawn at random and is put back into the urn along with k additional balls of the same colour as that of the ball drawn. A ball is again drawn at random. What is the probability that the ball drawn now is white?
(b) An unbiased die, with faces numbered $1,2,3,4,5,6$ is thrown n times and the list of n numbers showing up is noted. What is the probability that, among the numbers $1,2,3,4,5,6$ only three numbers appear in this list?

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) $\frac{75}{83}$

Let $S$ be the sample space of all $3 \times 3$ matrices with entries from the set $\{ 0,1 \}$. Let the events $E _ { 1 }$ and $E _ { 2 }$ be given by $$\begin{aligned} & E _ { 1 } = \{ A \in S : \operatorname { det } A = 0 \} \text { and } \\ & E _ { 2 } = \{ A \in S : \text { sum of entries of } A \text { is } 7 \} . \end{aligned}$$ If a matrix is chosen at random from $S$, then the conditional probability $P \left( E _ { 1 } \mid E _ { 2 } \right)$ equals $\_\_\_\_$

Two fair dice, each with faces numbered $1, 2, 3, 4, 5$ and $6$, are rolled together and the sum of the numbers on the faces is observed. This process is repeated till the sum is either a prime number or a perfect square. Suppose the sum turns out to be a perfect square before it turns out to be a prime number. If $p$ is the probability that this perfect square is an odd number, then the value of $14p$ is $\_\_\_\_$

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) $\frac { 3 } { 256 }$