LFM Stats And Pure

brazil-enem 2011 Q144 View

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

brazil-enem 2016 Q157 View

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) $6\%$

cmi-entrance 2022 QA2 4 marks View

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.

csat-suneung 2007 Q26 (Probability and Statistics) 3 marks View

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) $\frac { 15 } { 16 }$

csat-suneung 2007 Q28 4 marks View

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

csat-suneung 2008 Q7 3 marks View

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) $\frac { 5 } { 18 }$

csat-suneung 2009 Q16 4 marks View

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) $\frac { 7 } { 20 }$

csat-suneung 2011 Q5 3 marks View

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) $\frac { 1 } { 2 }$

csat-suneung 2012 Q10 3 marks View

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) $\frac { 3 } { 10 }$

csat-suneung 2014 Q5 3 marks View

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) $\frac { 7 } { 10 }$

csat-suneung 2014 Q7 3 marks View

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) $\frac { 1 } { 3 }$

csat-suneung 2016 Q5 3 marks View

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) $\frac { 3 } { 4 }$

csat-suneung 2017 Q4 3 marks View

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) $\frac { 1 } { 6 }$

csat-suneung 2018 Q4 3 marks View

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) $\frac { 2 } { 3 }$

csat-suneung 2018 Q10 3 marks View

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) $\frac { 2 } { 3 }$

csat-suneung 2019 Q27 4 marks View

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]

csat-suneung 2021 Q5 3 marks View

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) $\frac { 1 } { 6 }$

csat-suneung 2024 Q24 3 marks View

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) $\frac{7}{8}$

gaokao 2015 Q2 View

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

gaokao 2015 Q4 View

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

gaokao 2021 Q8 View

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

grandes-ecoles 2018 QII.4 View

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.

grandes-ecoles 2020 Q1 View

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

grandes-ecoles 2021 Q1b View

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.

grandes-ecoles 2021 Q3c View

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.

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