Bag A contains 5 cards with the numbers $1,2,3,4,5$ written on them, one each, and Bag B contains 5 cards with the numbers $6,7,8,9,10$ written on them, one each. One card is randomly drawn from each of the two bags A and B. When the sum of the two numbers on the drawn cards is odd, what is the probability that the number on the card drawn from Bag A is even? [3 points]
(1) $\frac { 5 } { 13 }$
(2) $\frac { 4 } { 13 }$
(3) $\frac { 3 } { 13 }$
(4) $\frac { 2 } { 13 }$
(5) $\frac { 1 } { 13 }$

Bag A contains 5 cards with the numbers $1,2,3,4,5$ written on them, one number per card, and bag B contains 5 cards with the numbers $6,7,8,9,10$ written on them, one number per card. One card is randomly drawn from each of the two bags A and B. When the sum of the two numbers on the drawn cards is odd, what is the probability that the number on the card drawn from bag A is even? [3 points]
(1) $\frac { 5 } { 13 }$
(2) $\frac { 4 } { 13 }$
(3) $\frac { 3 } { 13 }$
(4) $\frac { 2 } { 13 }$
(5) $\frac { 1 } { 13 }$

For two events $A$ and $B$, $\mathrm { P } ( A ) = \frac { 1 } { 2 } , \mathrm { P } \left( B ^ { C } \right) = \frac { 2 } { 3 }$, and $\mathrm { P } ( B \mid A ) = \frac { 1 } { 6 }$. What is the value of $\mathrm { P } \left( A ^ { C } \mid B \right)$? (Here, $A ^ { C }$ is the complement of $A$.) [3 points]
(1) $\frac { 1 } { 2 }$
(2) $\frac { 7 } { 12 }$
(3) $\frac { 2 } { 3 }$
(4) $\frac { 3 } { 4 }$
(5) $\frac { 5 } { 6 }$

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) $\frac { 9 } { 23 }$

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) $\frac { 9 } { 23 }$

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) $\frac { 1 } { 3 }$

For two events $A, B$, $$\mathrm{P}(A \cap B) = \frac{1}{8}, \quad \mathrm{P}\left(B^{C} \mid A\right) = 2\mathrm{P}(B \mid A)$$ what is the value of $\mathrm{P}(A)$? (Here, $B^{C}$ is the complement of $B$.) [3 points]
(1) $\frac{5}{12}$
(2) $\frac{3}{8}$
(3) $\frac{1}{3}$
(4) $\frac{7}{24}$
(5) $\frac{1}{4}$

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) $\frac { 9 } { 17 }$

For two events $A , B$, $$\mathrm { P } \left( A ^ { C } \cup B ^ { C } \right) = \frac { 4 } { 5 } , \quad \mathrm { P } \left( A \cap B ^ { C } \right) = \frac { 1 } { 4 }$$ what is the value of $\mathrm { P } \left( A ^ { C } \right)$? (Here, $A ^ { C }$ is the complement of $A$.) [3 points]
(1) $\frac { 1 } { 2 }$
(2) $\frac { 11 } { 20 }$
(3) $\frac { 3 } { 5 }$
(4) $\frac { 13 } { 20 }$
(5) $\frac { 7 } { 10 }$

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) $\frac { 3 } { 10 }$

Among 50 members of a marathon club who participated in a certain marathon, the number of members who completed the marathon and the number who withdrew are as follows. (Unit: persons)

Category	Male	Female
Completed	27	9
Withdrew	8	6

When one member is randomly selected from the participants and is found to be female, the probability that this member completed the marathon is $p$. Find the value of $100 p$. [3 points]

For two events $A$ and $B$, $$\mathrm { P } ( A ) = \frac { 1 } { 3 } , \quad \mathrm { P } ( A \cap B ) = \frac { 1 } { 8 }$$ what is the value of $\mathrm { P } \left( B ^ { C } \mid A \right)$? (Here, $B ^ { C }$ is the complement of $B$.) [4 points]
(1) $\frac { 11 } { 24 }$
(2) $\frac { 1 } { 2 }$
(3) $\frac { 13 } { 24 }$
(4) $\frac { 7 } { 12 }$
(5) $\frac { 5 } { 8 }$

For two events $A , B$, $$\mathrm { P } ( A ) = \frac { 2 } { 5 } , \quad \mathrm { P } ( B \mid A ) = \frac { 5 } { 6 }$$ what is the value of $\mathrm { P } ( A \cap B )$? [3 points]
(1) $\frac { 1 } { 3 }$
(2) $\frac { 4 } { 15 }$
(3) $\frac { 1 } { 5 }$
(4) $\frac { 2 } { 15 }$
(5) $\frac { 1 } { 15 }$

A company has a total of 60 employees, and each employee belongs to one of two departments, A or B. Department A has 20 employees and Department B has 40 employees. 50\% of the employees in Department A are women. 60\% of the women employees in the company belong to Department B. When one employee is randomly selected from the 60 employees and is found to belong to Department B, the probability that this employee is a woman is $p$. Find the value of $80p$. [4 points]

A survey was conducted on 500 students at a high school regarding their desire to visit regions A and B for cultural exploration. The results are as follows. (Unit: students)

Region B	Wish	Do not wish	Total
Wish	140	310	450
Do not wish	40	10	50
Total	180	320	500

When one student is randomly selected from this high school and is found to wish to visit region A, what is the probability that this student also wishes to visit region B? [3 points]
(1) $\frac { 19 } { 45 }$
(2) $\frac { 23 } { 45 }$
(3) $\frac { 3 } { 5 }$
(4) $\frac { 31 } { 45 }$
(5) $\frac { 7 } { 9 }$

A die is rolled twice. Given that the number 6 does not appear at all, what is the probability that the sum of the two numbers is a multiple of 4? [3 points]
(1) $\frac { 4 } { 25 }$
(2) $\frac { 1 } { 5 }$
(3) $\frac { 6 } { 25 }$
(4) $\frac { 7 } { 25 }$
(5) $\frac { 8 } { 25 }$

For two events $A$ and $B$, $A$ and $B ^ { C }$ are mutually exclusive events, and $$\mathrm { P } ( A ) = \frac { 1 } { 3 } , \mathrm { P } \left( A ^ { C } \cap B \right) = \frac { 1 } { 6 }$$ What is the value of $\mathrm { P } ( B )$? (Here, $A ^ { C }$ is the complement of $A$.) [3 points]
(1) $\frac { 5 } { 12 }$
(2) $\frac { 1 } { 2 }$
(3) $\frac { 7 } { 12 }$
(4) $\frac { 2 } { 3 }$
(5) $\frac { 3 } { 4 }$

For two events $A$ and $B$, $$\mathrm { P } ( B \mid A ) = \frac { 1 } { 4 } , \quad \mathrm { P } ( A \mid B ) = \frac { 1 } { 3 } , \quad \mathrm { P } ( A ) + \mathrm { P } ( B ) = \frac { 7 } { 10 }$$ what is the value of $\mathrm { P } ( A \cap B )$? [3 points]
(1) $\frac { 1 } { 7 }$
(2) $\frac { 1 } { 8 }$
(3) $\frac { 1 } { 9 }$
(4) $\frac { 1 } { 10 }$
(5) $\frac { 1 } { 11 }$

For two events $A , B$, $$\mathrm { P } ( A ) = \frac { 2 } { 5 } , \quad \mathrm { P } ( B \mid A ) = \frac { 1 } { 4 } , \quad \mathrm { P } ( A \cup B ) = 1$$ What is the value of $\mathrm { P } ( B )$? [3 points]
(1) $\frac { 7 } { 10 }$
(2) $\frac { 3 } { 4 }$
(3) $\frac { 4 } { 5 }$
(4) $\frac { 17 } { 20 }$
(5) $\frac { 9 } { 10 }$

There are 16 balls and six empty boxes with the natural numbers 1 through 6 written on them. A trial is performed using one die.
When the die is rolled and the result is $k$: If $k$ is odd, place 1 ball each in the boxes labeled $1, 3, 5$, and if $k$ is even, place 1 ball each in the boxes labeled with the divisors of $k$.
After repeating this trial 4 times, given that the sum of all balls in the six boxes is odd, what is the probability that the number of balls in the box labeled 3 is 1 more than the number of balls in the box labeled 2? [4 points]
(1) $\frac { 1 } { 8 }$
(2) $\frac { 3 } { 16 }$
(3) $\frac { 1 } { 4 }$
(4) $\frac { 5 } { 16 }$
(5) $\frac { 3 } { 8 }$

50 people registered for the soccer club, 60 people registered for the table tennis club, and 70 people registered for either the soccer or table tennis club. If a person is known to have registered for the soccer club, the probability that they also registered for the table tennis club is
A. $0.8$
B. $0.4$
C. $0.2$
D. $0.1$

Calculate $P _ { S } ( T )$ and $P _ { \bar { S } } ( T )$ and justify that the results of the experiment do not allow conclusions to be drawn about the effectiveness of the plant protection product against the tropical fungus.

In fact, the proportion of employees with a job ticket is different at the two locations; at location B, only half of the employees have a job ticket. Calculate the probability that a randomly selected employee of the automotive supplier who has a job ticket works at location A.

$49 \%$ of the packages contain clothing. Of the packages that are returns, $91 \%$ contain clothing.
A package is randomly selected. The following events are considered: $R$ : ``The package is a return.'' $K$ : ``The package contains clothing.'' Represent the described situation in a completely filled four-field table. For the case that the selected package is not a return, determine the probability that the package contains clothing.

155- A sample space consists of 5 outcomes $a, b, c, d, e$. If $P(a) = \dfrac{1}{4}$ and $P(\{a,b,c\}) = \dfrac{2}{3}$, then $P(\{b,c,e\} \mid \{a,b,c\})$ equals what?
(1) $\dfrac{5}{8}$ (2) $\dfrac{5}{12}$ (3) $\dfrac{5}{8}$ (4) $\dfrac{3}{4}$
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