Em um colégio, a probabilidade de um aluno ser aprovado em Matemática é 0,7 e a probabilidade de ser aprovado em Português é 0,8. Assumindo que as aprovações são eventos independentes, a probabilidade de um aluno ser aprovado em ambas as disciplinas é
(A) 0,14 (B) 0,24 (C) 0,56 (D) 0,75 (E) 0,90

A couple, both 30 years old, intends to take out a private pension plan. The insurance company researched, to define the value of the monthly contribution, estimates the probability that at least one of them will be alive in 50 years, based on population data, which indicate that 20\% of men and 30\% of women today will reach the age of 80.
What is this probability?
(A) $50\%$
(B) $44\%$
(C) $38\%$
(D) $25\%$
(E) $6\%$

Any two events $X$ and $Y$ are called mutually exclusive when the probability $P(X$ and $Y) = 0$ and they are called exhaustive when $P(X$ or $Y) = 1$. Suppose $A$ and $B$ are events and the probability of each of these two events is strictly between 0 and 1 (i.e., $0 < P(A) < 1$ and $0 < P(B) < 1$).
Statements
(5) $A$ and $B$ are mutually exclusive if and only if not $A$ and not $B$ are exhaustive. (6) $A$ and $B$ are independent if and only if not $A$ and not $B$ are independent. (7) $A$ and $B$ cannot be simultaneously independent and exhaustive. (8) $A$ and $B$ cannot be simultaneously mutually exclusive and exhaustive.

For two independent events $A$ and $B$, $$\mathrm { P } ( A \cap B ) = 2 \mathrm { P } \left( A \cap B ^ { c } \right) , \quad \mathrm { P } \left( A ^ { c } \cap B \right) = \frac { 1 } { 12 }$$ find the value of $\mathrm { P } ( A )$. (Given that $\mathrm { P } ( A ) \neq 0$.) [3 points]
(1) $\frac { 1 } { 2 }$
(2) $\frac { 5 } { 8 }$
(3) $\frac { 3 } { 4 }$
(4) $\frac { 7 } { 8 }$
(5) $\frac { 15 } { 16 }$

When 3 coins are tossed simultaneously, let $A$ be the event that at most 1 coin shows heads, and let $B$ be the event that all 3 coins show the same face. Which of the following statements in the given options are correct? [4 points] Given Options ㄱ. $\mathrm { P } ( A ) = \frac { 1 } { 2 }$ ㄴ. $\mathrm { P } ( A \cap B ) = \frac { 1 } { 8 }$ ㄷ. Events $A$ and $B$ are independent of each other.
(1) ㄱ
(2) ㄷ
(3) ㄱ, ㄴ
(4) ㄴ, ㄷ
(5) ㄱ, ㄴ, ㄷ

Two events $A , B$ are independent and $\mathrm { P } \left( A ^ { C } \right) = \mathrm { P } ( B ) = \frac { 1 } { 3 }$. What is the value of $\mathrm { P } ( A \cap B )$? (Here, $A ^ { C }$ is the complement of $A$.) [3 points]
(1) $\frac { 1 } { 18 }$
(2) $\frac { 1 } { 9 }$
(3) $\frac { 1 } { 6 }$
(4) $\frac { 2 } { 9 }$
(5) $\frac { 5 } { 18 }$

Bag A and Bag 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 Bag A, and Younghee draws one marble from Bag B. They check the numbers on the two marbles and do not put them back. This process 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) $\frac { 7 } { 20 }$

Two events $A$ and $B$ are mutually independent, and $$\mathrm { P } ( A ) = \frac { 2 } { 3 } , \mathrm { P } ( A \cap B ) = \mathrm { P } ( A ) - \mathrm { P } ( B )$$ What is the value of $\mathrm { P } ( B )$? [3 points]
(1) $\frac { 1 } { 10 }$
(2) $\frac { 1 } { 5 }$
(3) $\frac { 3 } { 10 }$
(4) $\frac { 2 } { 5 }$
(5) $\frac { 1 } { 2 }$

Two events $A$ and $B$ are independent, and
$$\mathrm { P } ( A \cup B ) = \frac { 1 } { 2 } , \quad \mathrm { P } ( A \mid B ) = \frac { 3 } { 8 }$$
What is the value of $\mathrm { P } \left( A \cap B ^ { C } \right)$? (where $B ^ { C }$ is the complement of $B$) [3 points]
(1) $\frac { 1 } { 10 }$
(2) $\frac { 3 } { 20 }$
(3) $\frac { 1 } { 5 }$
(4) $\frac { 1 } { 4 }$
(5) $\frac { 3 } { 10 }$

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) $\frac { 7 } { 10 }$

Two events $A , B$ are independent, and $\mathrm { P } ( A ) = \frac { 1 } { 3 } , \mathrm { P } ( B ) = \frac { 1 } { 3 }$. What is the value of $\mathrm { P } \left( A \cap B ^ { C } \right)$? (Here, $B ^ { C }$ is the complement of $B$.) [3 points]
(1) $\frac { 5 } { 27 }$
(2) $\frac { 2 } { 9 }$
(3) $\frac { 7 } { 27 }$
(4) $\frac { 8 } { 27 }$
(5) $\frac { 1 } { 3 }$

Two events $A$ and $B$ are independent, and $$\mathrm { P } \left( A ^ { C } \right) = \frac { 1 } { 4 } , \quad \mathrm { P } ( A \cap B ) = \frac { 1 } { 2 }$$ What is the value of $\mathrm { P } \left( B \mid A ^ { C } \right)$? (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) $\frac { 3 } { 4 }$

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

Two events $A$ and $B$ are mutually independent and $$\mathrm { P } ( A ) = \frac { 2 } { 3 } , \quad \mathrm { P } ( A \cup B ) = \frac { 5 } { 6 }$$ What is the value of $\mathrm { P } ( B )$? [3 points]
(1) $\frac { 1 } { 3 }$
(2) $\frac { 5 } { 12 }$
(3) $\frac { 1 } { 2 }$
(4) $\frac { 7 } { 12 }$
(5) $\frac { 2 } { 3 }$

Two events $A$ and $B$ are independent, and $$\mathrm { P } ( A ) = \frac { 2 } { 3 } , \quad \mathrm { P } ( A \cup B ) = \frac { 5 } { 6 }$$ Find the value of $\mathrm { P } ( B )$. [3 points]
(1) $\frac { 1 } { 3 }$
(2) $\frac { 5 } { 12 }$
(3) $\frac { 1 } { 2 }$
(4) $\frac { 7 } { 12 }$
(5) $\frac { 2 } { 3 }$

A die is rolled once. Let A be the event that an odd number appears, and for a natural number $m$ with $m \leq 6$, let B be the event that a divisor of $m$ appears. Find the sum of all values of $m$ such that the two events A and B are independent. [4 points]

Two events $A$ and $B$ are independent and $$\mathrm { P } ( A \mid B ) = \mathrm { P } ( B ) , \quad \mathrm { P } ( A \cap B ) = \frac { 1 } { 9 }$$ What is the value of $\mathrm { P } ( A )$? [3 points]
(1) $\frac { 7 } { 18 }$
(2) $\frac { 1 } { 3 }$
(3) $\frac { 5 } { 18 }$
(4) $\frac { 2 } { 9 }$
(5) $\frac { 1 } { 6 }$

Two events $A$ and $B$ are independent, and $$\mathrm{P}(A \cap B) = \frac{1}{4}, \quad \mathrm{P}(A^C) = 2\mathrm{P}(A)$$ Find the value of $\mathrm{P}(B)$. (Here, $A^C$ is the complement of $A$.) [3 points]
(1) $\frac{3}{8}$
(2) $\frac{1}{2}$
(3) $\frac{5}{8}$
(4) $\frac{3}{4}$
(5) $\frac{7}{8}$

2. Let $A$ and $B$ be two sets. Then ``$A \cap B = A$'' is ``$A \subseteq B$'' a
A. sufficient but not necessary condition
B. necessary but not sufficient condition
C. necessary and sufficient condition
D. neither sufficient nor necessary condition

4. Let $a , b$ be positive real numbers. Then ``$a > b > 1$'' is a \_\_\_\_ condition for ``$\log _ { 2 } a > \log _ { 2 } b > 0$''
(A) necessary and sufficient condition
(B) sufficient but not necessary condition
(C) necessary but not sufficient condition
(D) neither sufficient nor necessary condition

8. B

Solution: Let the probabilities of events A, B, C, D be $P ( a ) , P ( b ) , P ( c ) , P ( d )$ respectively. Then $P ( a ) = \frac { 1 } { 6 } , P ( b ) = \frac { 5 } { 36 } , P ( d ) = \frac { 1 } { 36 }$. The probability that A and C occur simultaneously is $P ( a c ) = 0$; the probability that A and D occur simultaneously is $P ( a d ) = 0$; the probability that B and C occur simultaneously is $P ( b c ) = \frac { 1 } { 36 }$; the probability that C and D occur simultaneously is $P ( c d ) = 0$. The condition $P ( x y ) = P ( x ) P ( y )$ is satisfied by option B.

II. Multiple Selection Questions

We introduce a uniformly distributed random variable $C : \Omega \rightarrow \mathcal{M}_{n}(\{-1,1\})$. For $\omega \in \Omega$, we denote by $C_{i,j}(\omega)$ the coefficients of the matrix $C(\omega)$.
Let $X = (x_{1}, \ldots, x_{n})$ and $Y = (y_{1}, \ldots, y_{n})$ be two arbitrary vectors in $\{-1,1\}^{n}$. Show that $\left(x_{i} y_{j} C_{i,j}\right)_{1 \leqslant i,j \leqslant n}$ is a family of $n^{2}$ random variables taking values in $\{-1,1\}$, independent and uniformly distributed.

Show that $S_n$ and $X_{n+1}$ are independent.

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.
Let $\left(\alpha_i\right)_{i \in \mathbb{N}^*}$ be a sequence of natural integers. Show that the events $$\left\{p_1^{\alpha_1} \mid X\right\}, \left\{p_2^{\alpha_2} \mid X\right\}, \ldots, \left\{p_k^{\alpha_k} \mid X\right\}, \ldots$$ are mutually independent.

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 $s$.
Deduce that the random variables $\nu_{p_1}(X), \ldots, \nu_{p_k}(X), \ldots$ are mutually independent.