For two events $A$ and $B$ in the sample space $S$, if they are mutually exclusive events, $A \cup B = S$, and $\mathrm { P } ( A ) = 2 \mathrm { P } ( B )$, what is the value of $\mathrm { P } ( A )$? [3 points]
(1) $\frac { 2 } { 3 }$
(2) $\frac { 1 } { 2 }$
(3) $\frac { 2 } { 5 }$
(4) $\frac { 1 } { 3 }$
(5) $\frac { 1 } { 4 }$

Two events $A$ and $B$ are mutually exclusive, and $$\mathrm { P } ( A ) = \mathrm { P } ( B ) , \quad \mathrm { P } ( A ) \mathrm { P } ( B ) = \frac { 1 } { 9 }$$ What is the value of $\mathrm { P } ( A \cup B )$? [3 points]
(1) $\frac { 1 } { 6 }$
(2) $\frac { 1 } { 3 }$
(3) $\frac { 1 } { 2 }$
(4) $\frac { 2 } { 3 }$
(5) $\frac { 5 } { 6 }$

A student named Chulsu participated in a design competition. Participants receive scores in two categories, and the possible scores in each category are one of three types shown in the table. The probability that Chulsu receives score A in each category is $\frac { 1 } { 2 }$, the probability of receiving score B is $\frac { 1 } { 3 }$, and the probability of receiving score C is $\frac { 1 } { 6 }$. When the event of receiving audience voting scores and the event of receiving judge scores are mutually independent, what is the probability that the sum of the two scores Chulsu receives is 70? [3 points]

Category	Score A	Score B	Score C
Audience Voting	40	30	20
Judges	50	40	30

(1) $\frac { 1 } { 3 }$
(2) $\frac { 11 } { 36 }$
(3) $\frac { 5 } { 18 }$
(4) $\frac { 1 } { 4 }$
(5) $\frac { 2 } { 9 }$

Chulsu participated in a certain design competition. Participants receive scores in two categories, and the possible scores in each category are one of three types shown in the table. The probability that Chulsu receives score A in each category is $\frac { 1 } { 2 }$, the probability of receiving score B is $\frac { 1 } { 3 }$, and the probability of receiving score C is $\frac { 1 } { 6 }$. When the event of receiving an audience vote score and the event of receiving a judge score are independent, what is the probability that the sum of the two scores Chulsu receives is 70? [3 points]

Category	Score A	Score B	Score C
Audience Vote	40	30	20
Judge	50	40	30

(1) $\frac { 1 } { 3 }$
(2) $\frac { 11 } { 36 }$
(3) $\frac { 5 } { 18 }$
(4) $\frac { 1 } { 4 }$
(5) $\frac { 2 } { 9 }$

For two events $A , B$, $A ^ { C }$ and $B$ are mutually exclusive events, and $$\mathrm { P } ( A ) = 2 \mathrm { P } ( B ) = \frac { 3 } { 5 }$$ When this condition is satisfied, what is the value of $\mathrm { P } \left( A \cap B ^ { C } \right)$? (Here, $A ^ { C }$ is the complement of $A$.) [3 points]
(1) $\frac { 3 } { 20 }$
(2) $\frac { 1 } { 5 }$
(3) $\frac { 1 } { 4 }$
(4) $\frac { 3 } { 10 }$
(5) $\frac { 7 } { 20 }$

For two events $A$ and $B$, $$\mathrm { P } ( A \cap B ) = \frac { 1 } { 8 } , \mathrm { P } \left( A \cap B ^ { C } \right) = \frac { 3 } { 16 }$$ What is the value of $\mathrm { P } ( A )$? (Here, $B ^ { C }$ is the complement of $B$.) [3 points]
(1) $\frac { 3 } { 16 }$
(2) $\frac { 7 } { 32 }$
(3) $\frac { 1 } { 4 }$
(4) $\frac { 9 } { 32 }$
(5) $\frac { 5 } { 16 }$

A box contains 5 white masks and 9 black masks. When 3 masks are randomly drawn simultaneously from the box, what is the probability that at least one of the 3 masks is white? [3 points]
(1) $\frac { 8 } { 13 }$
(2) $\frac { 17 } { 26 }$
(3) $\frac { 9 } { 13 }$
(4) $\frac { 19 } { 26 }$
(5) $\frac { 10 } { 13 }$

Let $s > 1$ be a real number and let $X$ be a random variable taking values in $\mathbb{N}^*$ following the zeta distribution with parameter $s$. If $n \in \mathbb{N}^*$, we denote $\{n \mid X\}$ the event ``$n$ divides $X$'' and $\{n \nmid X\}$ the complementary event.
Calculate $P(n \mid X)$ for $n \in \mathbb{N}^*$.

Let $G _ { 0 } = \left( S _ { 0 } , A _ { 0 } \right)$ be a particular fixed graph with $s _ { 0 } = s _ { G _ { 0 } }$, $a _ { 0 } = a _ { G _ { 0 } }$, $s_0 \geq 2$, $a_0 \geq 1$. For a graph $H = \left( S _ { H } , A _ { H } \right)$ with $S _ { H } \subset \llbracket 1 , n \rrbracket$, the random variable $X_H$ is defined by: $$\forall G \in \Omega _ { n } \quad X _ { H } ( G ) = \begin{cases} 1 & \text { if } H \subset G \\ 0 & \text { otherwise } \end{cases}$$ Show that $$\mathbf { E } \left( X _ { H } \right) = p _ { n } ^ { a _ { H } } .$$

We fix $n \in \mathbf { N } ^ { * }$ and draw successively and with replacement two integers $p$ and $q$ according to a uniform distribution on $\llbracket 1 , n \rrbracket$. The event $E_n$ is defined as "We obtain $(p,q) \in E_1 \cup E_2 \cup E_3$".
Using the result $\mathbf{P}(A_n \cup B_n) \sim \dfrac{\ln n}{n}$ as $n \to +\infty$, deduce $$\lim _ { n \rightarrow + \infty } \mathbf { P } \left( E _ { n } \right) .$$

Four persons independently solve a certain problem correctly with probabilities $\frac { 1 } { 2 } , \frac { 3 } { 4 } , \frac { 1 } { 4 } , \frac { 1 } { 8 }$. Then the probability that the problem is solved correctly by at least one of them is
(A) $\frac { 235 } { 256 }$
(B) $\frac { 21 } { 256 }$
(C) $\frac { 3 } { 256 }$
(D) $\frac { 253 } { 256 }$

Let $E$, $F$ and $G$ be three events having probabilities $$P(E) = \frac{1}{8}, \quad P(F) = \frac{1}{6}, \quad P(G) = \frac{1}{4},$$ and let $P(E \cap F \cap G) = \frac{1}{10}$.
For any event $H$, if $P(H^c)$ denotes its complement, then which of the following statements is(are) TRUE?
(A) $P(E \cap F \cap G^c) \leq \frac{1}{40}$
(B) $P(E^c \cap F \cap G) \leq \frac{1}{15}$
(C) $P(E \cup F \cup G) \leq \frac{13}{24}$
(D) $P(E^c \cap F^c \cap G^c) \leq \frac{5}{12}$

Consider an experiment of tossing a coin repeatedly until the outcomes of two consecutive tosses are same. If the probability of a random toss resulting in head is $\frac { 1 } { 3 }$, then the probability that the experiment stops with head is
(A) $\frac { 1 } { 3 }$
(B) $\frac { 5 } { 21 }$
(C) $\frac { 4 } { 21 }$
(D) $\frac { 2 } { 7 }$

If $A$ and $B$ are two events such that $P ( A \cup B ) = P ( A \cap B )$, then the incorrect statement amongst the following statements is:
(1) $P ( A ) + P ( B ) = 1$
(2) $P \left( A \cap B ^ { \prime } \right) = 0$
(3) $A \& B$ are equally likely
(4) $P \left( A ^ { \prime } \cap B \right) = 0$

For three events $A$, $B$ and $C$, $P(\text{Exactly one of } A \text{ or } B \text{ occurs}) = P(\text{Exactly one of } B \text{ or } C \text{ occurs}) = P(\text{Exactly one of } C \text{ or } A \text{ occurs}) = \dfrac{1}{4}$ and $P(\text{All the three events occur simultaneously}) = \dfrac{1}{16}$. Then the probability that at least one of the events occurs, is:
(1) $\dfrac{7}{32}$
(2) $\dfrac{7}{16}$
(3) $\dfrac{1}{64}$
(4) $\dfrac{3}{16}$

An unbiased coin is tossed eight times. The probability of obtaining at least one head and at least one tail is:
(1) $\frac { 127 } { 128 }$
(2) $\frac { 63 } { 64 }$
(3) $\frac { 255 } { 256 }$
(4) $\frac { 1 } { 2 }$

In a class of 60 students, 40 opted for NCC, 30 opted for NSS and 20 opted for both NCC and NSS. If one of these students is selected at random, then the probability that the student selected has opted neither for NCC nor for NSS is :
(1) $\frac { 1 } { 6 }$
(2) $\frac { 5 } { 6 }$
(3) $\frac { 1 } { 3 }$
(4) $\frac { 2 } { 3 }$

Four persons can hit a target correctly with probabilities $\frac { 1 } { 2 } , \frac { 1 } { 3 } , \frac { 1 } { 4 }$ and $\frac { 1 } { 8 }$ respectively. If all hit at the target independently, then the probability that the target would be hit, is
(1) $\frac { 25 } { 192 }$
(2) $\frac { 7 } { 32 }$
(3) $\frac { 1 } { 192 }$
(4) $\frac { 25 } { 32 }$

Two newspapers $A$ and $B$ are published in a city. It is known that $25 \%$ of the city population reads $A$ and $20 \%$ reads $B$ while $8 \%$ reads both $A$ and $B$. Further, 30\% of those who read $A$ but not $B$ look into advertisements and $40 \%$ of those who read $B$ but not $A$ also look into advertisements, while $50 \%$ of those who read both $A$ and $B$ look into advertisements. Then the percentage of the population who look into advertisements is:
(1) 13.5
(2) 12.8
(3) 13.9
(4) 13

Let $A , B$ and $C$ be three events such that the probability that exactly one of $A$ and $B$ occurs is $( 1 - k )$, the probability that exactly one of $B$ and $C$ occurs is $( 1 - 2k )$, the probability that exactly one of $C$ and $A$ occurs is $( 1 - k )$ and the probability of all $A , B$ and $C$ occur simultaneously is $k ^ { 2 }$, where $0 < k < 1$. Then the probability that at least one of $A , B$ and $C$ occur is:
(1) greater than $\frac { 1 } { 8 }$ but less than $\frac { 1 } { 4 }$
(2) greater than $\frac { 1 } { 2 }$
(3) greater than $\frac { 1 } { 4 }$ but less than $\frac { 1 } { 2 }$
(4) exactly equal to $\frac { 1 } { 2 }$

Let $E _ { 1 } , E _ { 2 } , E _ { 3 }$ be three mutually exclusive events such that $P \left( E _ { 1 } \right) = \frac { 2 + 3 p } { 6 } , P \left( E _ { 2 } \right) = \frac { 2 - p } { 8 }$ and $P \left( E _ { 3 } \right) = \frac { 1 - p } { 2 }$. If the maximum and minimum values of $p$ are $p _ { 1 }$ and $p _ { 2 }$ then $\left( p _ { 1 } + p _ { 2 } \right)$ is equal to:
(1) $\frac { 2 } { 3 }$
(2) $\frac { 5 } { 3 }$
(3) $\frac { 5 } { 4 }$
(4) 1

A village mayor election has two polling stations. The proportion of valid votes received by the two candidates at each polling station is shown in the following table (invalid votes are not counted):

	Candidate A	Candidate B
First Polling Station	$40 \%$	$60 \%$
Second Polling Station	$55 \%$	$45 \%$

Assume the number of valid votes at the first and second polling stations are $x$ and $y$ respectively (where $x > 0 , y > 0$), and the candidate with the higher total votes wins. Based on the above table, select the correct options.
(1) When the total number of valid votes $x + y$ is known, the winner can be determined
(2) When the ratio $x : y$ is less than $\frac { 1 } { 2 }$, the winner can be determined
(3) When $x > y$, the winner can be determined
(4) When Candidate A's valid votes at the first polling station exceed those at the second polling station, the winner can be determined
(5) When Candidate B's valid votes at the second polling station exceed those at the first polling station, the winner can be determined

There are two parking lots next to a scenic spot. Assume that on a certain day, the probability that either parking lot has no available spaces is 0.7, and whether the two parking lots have available spaces is independent. If a car arrives at these two parking lots on that day, the probability that at least one parking lot has available spaces is 0.(13--1)(13--2).