Bag $A$ contains 3 white, 7 red balls and bag $B$ contains 3 white, 2 red balls. One bag is selected at random and a ball is drawn from it. The probability of drawing the ball from the bag $A$, if the ball drawn is white, is:
(1) $\frac{1}{4}$
(2) $\frac{1}{9}$
(3) $\frac{1}{3}$
(4) $\frac{3}{10}$

There are three bags $X , Y$ and $Z$. Bag $X$ contains 5 one-rupee coins and 4 five-rupee coins; Bag $Y$ contains 4 one-rupee coins and 5 five-rupee coins and Bag Z contains 3 one-rupee coins and 6 five-rupee coins. A bag is selected at random and a coin drawn from it at random is found to be a one-rupee coin. Then the probability, that it came from bag Y , is : (1) $\frac { 1 } { 4 }$ (2) $\frac { 1 } { 2 }$ (3) $\frac { 5 } { 12 }$ (4) $\frac { 1 } { 3 }$

Two balls are selected at random one by one without replacement from a bag containing 4 white and 6 black balls. If the probability that the first selected ball is black, given that the second selected ball is also black, is $\frac { m } { n }$, where $\operatorname { gcd } ( m , n ) = 1$, then $m + n$ is equal to:
(1) 4
(2) 14
(3) 13
(4) 11

If $A$ and $B$ are two events such that $P ( A \cap B ) = 0.1$, and $P ( A \mid B )$ and $P ( B \mid A )$ are the roots of the equation $12 x ^ { 2 } - 7 x + 1 = 0$, then the value of $\frac { \mathrm { P } ( \overline { \mathrm { A } } \cup \overline { \mathrm { B } } ) } { \mathrm { P } ( \overline { \mathrm { A } } \cap \overline { \mathrm { B } } ) }$ is :
(1) $\frac { 4 } { 3 }$
(2) $\frac { 7 } { 4 }$
(3) $\frac { 5 } { 3 }$
(4) $\frac { 9 } { 4 }$

Bag $B _ { 1 }$ contains 6 white and 4 blue balls, Bag $B _ { 2 }$ contains 4 white and 6 blue balls, and Bag $B _ { 3 }$ contains 5 white and 5 blue balls. One of the bags is selected at random and a ball is drawn from it. If the ball is white, then the probability, that the ball is drawn from Bag $B _ { 2 }$, is :
(1) $\frac { 4 } { 15 }$
(2) $\frac { 1 } { 3 }$
(3) $\frac { 2 } { 5 }$
(4) $\frac { 2 } { 3 }$

$A$ and $B$ alternately throw a pair of dice. $A$ wins if he throws a sum of 5 before $B$ throws a sum of 8, and $B$ wins if he throws a sum of 8 before $A$ throws a sum of 5. The probability that $A$ wins if $A$ makes the first throw, is
(1) $\frac{8}{17}$
(2) $\frac{9}{19}$
(3) $\frac{9}{17}$
(4) $\frac{8}{19}$

A research team uses a certain rapid test reagent to understand the proportion of organisms in a protected area whose body toxin accumulation exceeds the standard due to environmental pollution. The test result of this reagent shows only two colors: red and yellow. Based on past experience, it is known that: if body toxin accumulation exceeds the standard, after testing with this reagent, $75\%$ shows red; if body toxin accumulation does not exceed the standard, after testing with this reagent, $95\%$ shows yellow. It is known that for a certain type of organism in this protected area, $7.8\%$ of the test results show red. Assuming the actual proportion of this type of organism with body toxin accumulation exceeding the standard is $p\%$ , select the correct option.
(1) $1 \leq p < 3$
(2) $3 \leq p < 5$
(3) $5 \leq p < 7$
(4) $7 \leq p < 9$
(5) $9 \leq p < 11$

A psychologist conducted an experiment with 1000 subjects in a dark room, where each subject had to observe and identify three digit cards: 6, 8, and 9. The probability of mistaking the actual digit for another digit is shown in the following table:

\backslashbox{Actual Digit}{Seen as}	6	8	9	Other
6	0.4	0.3	0.2	0.1
8	0.3	0.4	0.1	0.2
9	0.2	0.2	0.5	0.1

For example: The actual digit 6 is seen as 6, 8, 9 with probabilities 0.4, 0.3, 0.2 respectively, and is seen as another digit with probability 0.1. Based on the above experimental results, select the correct options.
(1) If the actual digit is 8, then there is at least a 50\% chance it will be seen as 8
(2) If the actual digit is 6, then there is a 60\% chance it will be seen as not 6
(3) Among the three digits 6, 8, 9, the digit 9 has the lowest probability of being misidentified
(4) If the digit seen is 6, then the probability that it is actually 6 is less than 50\%
(5) If the digit seen is 9, then the probability that it is actually 9 is greater than $\frac { 2 } { 3 }$

It is known that 30\% of the population in a certain region is infected with a certain infectious disease. For a rapid screening test of the disease, there are two results: positive or negative. The test has an 80\% probability of identifying an infected person as positive and a 60\% probability of identifying an uninfected person as negative. To reduce the situation where the test incorrectly identifies an infected person as negative, experts recommend three consecutive tests. If $P$ is the probability that an infected person is among those who test negative in a single test, and $P'$ is the probability that an infected person is among those who test negative in all three consecutive tests, what is $\frac { P } { P' }$ closest to?
(1) 7
(2) 8
(3) 9
(4) 10
(5) 11

According to a certain country's investigation of missing light aircraft: 70\% of missing light aircraft are eventually found. Among the aircraft that are found, 60\% have emergency locator transmitters installed; among the missing aircraft that are not found, 90\% do not have emergency locator transmitters installed. Emergency locator transmitters send signals when the aircraft crashes, allowing rescue personnel to locate it. A light aircraft is now missing. If it is known that the aircraft has an emergency locator transmitter installed, the probability that it will be found is (15--1)(15--2).

A bag contains 100 balls numbered $1, 2, \ldots, 100$ respectively. A person randomly draws one ball from the bag, and each ball has an equal probability of being drawn. Under which of the following conditions is the conditional probability that the person draws ball number 7 the largest?
(1) The number of the ball drawn is odd (2) The number of the ball drawn is prime (3) The number of the ball drawn is a multiple of 7 (4) The number of the ball drawn is not a multiple of 5 (5) The number of the ball drawn is less than 10

All senior high school students at a certain school have taken either Mathematics A or Mathematics B on the scholastic aptitude test. Among these students, those taking only Mathematics A account for $\frac { 3 } { 10 }$ of all senior high school students. Among students taking Mathematics A, $\frac { 5 } { 8 }$ also took Mathematics B. What is the proportion of students taking only Mathematics B among all students at the school taking Mathematics B? (Express as a fraction in lowest terms)

There are two fair six-sided dice A and B: The numbers on A are $1, 2, 5, 6, 7, 9$, The numbers on B are $1, 3, 4, 5, 6, 9$. The relationship between the numbers on A and B is recorded in the table below. For example: if the numbers on A and B are 5 and 3 respectively, it is recorded as ``A wins''; if both A and B show 5, it is recorded as ``tie''.

\multirow{2}{*}{}		\multicolumn{6}{|c|}{A}
	Number	1	2	5	6	7	9
\multirow{6}{*}{B}	1	Tie	A wins	A wins	A wins	A wins	A wins
	3	B wins	B wins	A wins	A wins	A wins	A wins
	4	B wins	B wins	A wins	A wins	A wins	A wins
	5	B wins	B wins	Tie	A wins	A wins	A wins
	6	B wins	B wins	B wins	Tie	A wins	A wins
	9	B wins	B wins	B wins	B wins	B wins	Tie

If a person rolls both dice A and B simultaneously, what is the probability that B shows 6 given that A's number is greater than B's number?
(1) $\frac { 1 } { 6 }$
(2) $\frac { 1 } { 9 }$
(3) $\frac { 1 } { 16 }$
(4) $\frac { 1 } { 18 }$
(5) $\frac { 1 } { 32 }$

Problem 6

A product factory manufactures 2 types of products: product-I and product-II. Part-A is necessary for product-I, and both part-$A$ and part-$B$ are necessary for product-II. There are parts that have standard quality and parts that do not have standard quality among part-$A$ and part-$B$. All parts are delivered from the part factory to the product factory, but there is no quality check of any part. The qualities of part-$A$ and part-$B$ are independent, and they will not affect each other. The probabilities that part-$A$ and part-$B$ have standard quality are $a$ and $b$, respectively.
A final quality inspection is made in the product factory for product-I and for product-II before shipment. The inspection judges whether the quality of each product meets the standard or not. The inspections will not affect each other. The product inspection is not perfect: namely, products that have standard quality pass the product inspection as acceptable with the probability $x$. The products that do not have standard quality pass the product inspection as acceptable with the probability $y$.
Answer the following questions: I. A product-I is randomly sampled and inspected once. Here, the probability that product-I can be manufactured with standard quality is defined as follows:

• The probability that product-I has standard quality is $c$ if part-$A$ has standard quality.
• Product-I will never have standard quality if part-A does not have standard quality.

1. Show the probability that the selected product-I passes the product inspection as acceptable.
2. Show the probability that the selected product-I actually has standard quality after it has passed the product inspection as acceptable.

II. A product-II is randomly sampled and inspected $n$ times. Here, the probability that product-II can be manufactured with standard quality is defined as follows:

• The probability that product-II has standard quality is $c$ if both part-$A$ and part-$B$ have standard quality.
• The probability that product-II has standard quality is $d$ if only either part-$A$ or part-$B$ has standard quality.
• Product-II will never have standard quality if both part-$A$ and part-$B$ do not have standard quality.

1. Show the probability that the selected product-II has standard quality.
2. Show the probability that the selected product-II actually has standard quality after it has passed all product inspections (i.e., $n$ times) as acceptable.

A bag contains 10 balls numbered from 1 to 10.
Given that the sum of the numbers on two balls randomly drawn from the bag is 15, what is the probability that ball number 7 was drawn?
A) $\frac { 2 } { 3 }$
B) $\frac { 2 } { 5 }$
C) $\frac { 2 } { 7 }$
D) $\frac { 1 } { 2 }$
E) $\frac { 1 } { 3 }$

On a table, there are three marbles in total: one red, one blue, and one yellow. These marbles are placed in bags A, B, and C with one marble in each bag, and p: ``There is no red marble in bag A.'' q: ``There is a blue marble in bag B.'' r: ``There is no yellow marble in bag C.'' propositions are given.
$$p \wedge ( q \vee r ) ^ { \prime \prime }$$
Given that the proposition is true; what are the colors of the marbles in bags A, B and C respectively?
A) Red - Blue - Yellow
B) Blue - Red - Yellow
C) Blue - Yellow - Red
D) Yellow - Red - Blue
E) Yellow - Blue - Red

Regarding the subsets $A, B$ and $C$ of the set of natural numbers, the propositions
$$\begin{aligned} & p : 9 \in A \cup B \\ & q : 9 \in A \cap C \\ & r : 9 \notin C \end{aligned}$$
are given. Given that the proposition $(p \Rightarrow q)' \wedge r'$ is true, which of the following statements are true?
I. $9 \in A$ II. $9 \in B$ III. $9 \in C$
A) Only I B) Only III C) I and II D) II and III E) I, II and III