Question 154
Uma urna contém 5 bolas vermelhas, 3 bolas azuis e 2 bolas verdes. Retira-se uma bola ao acaso. A probabilidade de que a bola retirada seja vermelha ou verde é
(A) $\dfrac{1}{10}$ (B) $\dfrac{1}{5}$ (C) $\dfrac{7}{10}$ (D) $\dfrac{3}{5}$ (E) $\dfrac{7}{10}$

Uma urna contém 5 bolas vermelhas, 3 bolas azuis e 2 bolas verdes. Retira-se uma bola ao acaso. A probabilidade de que a bola retirada seja azul ou verde é
(A) $\dfrac{1}{10}$ (B) $\dfrac{1}{5}$ (C) $\dfrac{3}{10}$ (D) $\dfrac{1}{2}$ (E) $\dfrac{3}{5}$

José, Paulo, and Antônio are playing fair dice, on which, on each of the six faces, there is a number from 1 to 6. Each of them will roll two dice simultaneously. José believes that, after rolling his dice, the numbers on the faces facing up will give him a sum equal to 7. Paulo believes that his sum will be equal to 4, and Antônio believes that his sum will be equal to 8.
With this choice, who has the greatest probability of getting their respective sum correct is
(A) Antônio, since his sum is the largest of all the chosen ones.
(B) José and Antônio, since there are 6 possibilities for both José's choice and Antônio's choice, and there are only 4 possibilities for Paulo's choice.
(C) José and Antônio, since there are 3 possibilities for both José's choice and Antônio's choice, and there are only 2 possibilities for Paulo's choice.
(D) José, since there are 6 possibilities to form his sum, 5 possibilities to form Antônio's sum, and only 3 possibilities to form Paulo's sum.
(E) Paulo, since his sum is the smallest of all.

In a certain theater, the seats are divided into sectors. The figure presents the view of sector 3 of this theater, in which the dark chairs are reserved and the light ones have not been sold.
The ratio that represents the quantity of reserved chairs in sector 3 in relation to the total number of chairs in that same sector is
(A) $\frac{17}{70}$ (B) $\frac{17}{53}$ (C) $\frac{53}{70}$ (D) $\frac{53}{17}$ (E) $\frac{70}{17}$

A store monitored the number of buyers of two products, A and B, during the months of January, February and March 2012. With this, it obtained this graph.
The store will draw a prize among the buyers of product A and another prize among the buyers of product B.
What is the probability that both winners made their purchases in February 2012?
(A) $\frac{1}{20}$ (B) $\frac{3}{242}$ (C) $\frac{5}{22}$ (D) $\frac{6}{25}$ (E) $\frac{7}{15}$

A screw factory has two machines, I and II, for the production of a certain type of screw.
In September, machine I produced $\frac{54}{100}$ of the total screws produced by the factory. Of the screws produced by this machine, $\frac{25}{1000}$ were defective. In turn, $\frac{38}{1000}$ of the screws produced in the same month by machine II were defective.
The combined performance of the two machines is classified according to the table, in which $P$ indicates the probability of a randomly chosen screw being defective.
$$\begin{aligned} 0 \leq P < \frac{2}{100} & \quad \text{Excellent} \\ \frac{2}{100} \leq P < \frac{4}{100} & \quad \text{Good} \\ \frac{4}{100} \leq P < \frac{6}{100} & \quad \text{Fair} \\ \frac{6}{100} \leq P < \frac{8}{100} & \quad \text{Poor} \\ \frac{8}{100} \leq P \leq 1 & \quad \text{Very Poor} \end{aligned}$$
The combined performance of these machines in September can be classified as
(A) excellent. (B) good. (C) fair. (D) poor. (E) very poor.

QUESTION 151
A bag contains 5 red balls, 3 blue balls, and 2 green balls. The probability of randomly drawing a blue ball is
(A) $\frac{1}{5}$
(B) $\frac{3}{10}$
(C) $\frac{2}{5}$
(D) $\frac{1}{2}$
(E) $\frac{3}{5}$

A box contains a $\mathrm{R}\$ 5.00$ bill, a $\mathrm{R}\$ 20.00$ bill, and two $\mathrm{R}\$ 50.00$ bills of different designs. A bill is randomly drawn from this box, its value is noted, and the bill is returned to the box. Then, the previous procedure is repeated.
The probability that the sum of the noted values is at least equal to $\mathrm{R}\$ 55.00$ is
(A) $\frac{1}{2}$
(B) $\frac{1}{4}$
(C) $\frac{3}{4}$
(D) $\frac{2}{9}$
(E) $\frac{5}{9}$

The figure illustrates a game of Minesweeper, the game present in practically every personal computer. Four squares on a $16 \times 16$ board were opened, and the numbers on their faces indicate how many of their 8 neighbors contain mines (to be avoided). The number 40 in the lower right corner is the total number of mines on the board, whose positions were chosen at random, uniformly, before opening any square.
In his next move, the player must choose among the squares marked with the letters $P, Q, R, S$ and $T$ one to open, and should choose the one with the lowest probability of containing a mine.
The player should open the square marked with the letter
(A) $P$.
(B) $Q$.
(C) $R$.
(D) $S$.
(E) $T$.

A box contains 5 red balls, 3 blue balls, and 2 green balls. One ball is drawn at random from the box.
What is the probability that the drawn ball is blue?
(A) $\dfrac{1}{10}$
(B) $\dfrac{1}{5}$
(C) $\dfrac{3}{10}$
(D) $\dfrac{2}{5}$
(E) $\dfrac{1}{2}$

Four candidates presented themselves to take the exam of a competition. Before starting the exam, the cell phones of the four candidates were collected by the proctor, who stored them, each one, inside a black envelope. At the end of the exam, the proctor returned the four envelopes with the cell phones to the four candidates, in a random manner, since he had not identified the envelopes.
The probability that all candidates received back the envelopes with their respective cell phones is
(A) $\dfrac{1}{2}$
(B) $\dfrac{1}{10}$
(C) $\dfrac{1}{16}$
(D) $\dfrac{1}{24}$
(E) $\dfrac{1}{256}$

A magazine report addressed the use of social networks by Brazilian internet users. Some of the data collected by the report are presented in the infographic.
According to the infographic data, when randomly selecting a Brazilian internet user in the period to which the report refers, the probability that he is a man who accesses some social network is
(A) $\dfrac{30}{90}$
(B) $\dfrac{36}{100}$
(C) $\dfrac{40}{100}$
(D) $\dfrac{40}{90}$
(E) $\dfrac{46}{90}$

Positive integers $a$ and $b$, possibly equal, are chosen randomly from among the divisors of 400. The numbers $a, b$ are chosen independently, each divisor being equally likely to be chosen. Find the probability that $\gcd(a, b) = 1$ and $\text{lcm}(a, b) = 400$.

You are given an $8 \times 8$ chessboard. If two distinct squares are chosen uniformly at random find the probability that two rooks placed on these squares attack each other. Recall that a rook can move either horizontally or vertically, in a straight line.

Suppose $A$, $B$ and $C$ are three events and $P ( A ) = a , P ( B ) = b , P ( C ) = c$ are known. Let $P ( A \cup B \cup C ) = p$. The statements below are about whether we can find the value of $p$ if we know some additional information. (Note: $\cup$ is the same as OR. Similarly $\cap$ is the same as AND.)
Statements
(33) We can find the value of $p$ if we know that at least one of $a , b , c$ is 1. (34) We can find the value of $p$ if we know that at least one of $a , b , c$ is 0. (35) We can find the value of $p$ if we know that any two of $A , B$ and $C$ are mutually exclusive. (36) We can find the value of $p$ if we know that any two of $A , B$ and $C$ are independent and we know the value of $P ( A \cap B \cap C )$.

There are four people of different heights. When they stand in a line, what is the probability that the third person from the front is shorter than both of their neighbors? [3 points]
(1) $\frac { 1 } { 3 }$
(2) $\frac { 1 } { 2 }$
(3) $\frac { 3 } { 5 }$
(4) $\frac { 2 } { 3 }$
(5) $\frac { 3 } { 4 }$

There are four people of different heights. When they are arranged in a line, what is the probability that the third person from the front is shorter than the two people adjacent to him? [3 points]
(1) $\frac { 1 } { 3 }$
(2) $\frac { 1 } { 2 }$
(3) $\frac { 3 } { 5 }$
(4) $\frac { 2 } { 3 }$
(5) $\frac { 3 } { 4 }$

When two dice are rolled simultaneously, what is the probability that the number on one die is a multiple of the number on the other die? [4 points]
(1) $\frac { 7 } { 18 }$
(2) $\frac { 1 } { 2 }$
(3) $\frac { 11 } { 18 }$
(4) $\frac { 13 } { 18 }$
(5) $\frac { 5 } { 6 }$

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) $\frac { 1 } { 4 }$

Among $3n$ cards labeled with the numbers $1,2,3 , \cdots , 3 n$ ($n$ is a natural number), two cards are randomly drawn, and the two numbers on them are denoted as $a , b$ ($a < b$) respectively. Let $\mathrm { P } _ { n }$ be the probability that $3 a < b$. The following is the process of finding $\lim _ { n \rightarrow \infty } \mathrm { P } _ { n }$. The number of ways to draw 2 cards from $3 n$ cards is ${ } _ { 3 n } \mathrm { C } _ { 2 }$. When $3 a < b$, we have $b \leqq 3 n$, so $1 \leqq a < n$. Therefore, if $a = k$, the number of cases of $b$ satisfying $3 a < b$ is (A), so $$\mathrm { P } _ { n } = \frac { \text { (B) } } { { } _ { 3 n } \mathrm { C } _ { 2 } } \text { . }$$ Therefore, $\lim _ { n \rightarrow \infty } \mathrm { P } _ { n } =$ (C).
What are the correct values for (A), (B), and (C)? [4 points]

	(A)	(B)	(C)
(1)	$3 ( n - k )$	$\frac { 3 } { 2 } n ( n - 1 )$	$\frac { 1 } { 3 }$
(2)	$3 ( n - k )$	$\frac { 3 } { 2 } n ( n - 1 )$	$\frac { 2 } { 3 }$
(3)	$3 ( n - k )$	$3 n ( n - 1 )$	$\frac { 2 } { 3 }$
(4)	$3 ( n - k + 1 )$	$3 n ( n - 1 )$	$\frac { 1 } { 3 }$
(5)	$3 ( n - k + 1 )$	$3 n ( n - 1 )$	$\frac { 2 } { 3 }$

Six students $\mathrm { A } , \mathrm { B } , \mathrm { C } , \mathrm { D } , \mathrm { E } , \mathrm { F }$ are to be randomly paired into 3 groups of 2. What is the probability that A and B are in the same group and C and D are in different groups? [4 points]
(1) $\frac { 1 } { 15 }$
(2) $\frac { 1 } { 10 }$
(3) $\frac { 2 } { 15 }$
(4) $\frac { 1 } { 6 }$
(5) $\frac { 1 } { 5 }$

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

In information theory, when an event $E$ occurs, the information content $I ( E )$ of event $E$ is defined as follows:
$$I ( E ) = - \log _ { 2 } \mathrm { P } ( E )$$
Which of the following statements in $\langle$Remarks$\rangle$ are correct? (Note: The probability $\mathrm { P } ( E )$ of event $E$ is positive, and the unit of information content is bits.) [4 points]
$\langle$Remarks$\rangle$ ㄱ. If event $E$ is rolling an odd number on a single die, then $I ( E ) = 1$. ㄴ. If two events $A$ and $B$ are independent and $\mathrm { P } ( A \cap B ) > 0$, then $I ( A \cap B ) = I ( A ) + I ( B )$. ㄷ. For two events $A$ and $B$ with $\mathrm { P } ( A ) > 0$ and $\mathrm { P } ( B ) > 0$, we have $2 I ( A \cup B ) \leqq I ( A ) + I ( B )$.
(1) ㄱ
(2) ㄱ, ㄴ
(3) ㄱ, ㄷ
(4) ㄴ, ㄷ
(5) ㄱ, ㄴ, ㄷ

In information theory, when an event $E$ occurs, the information content $I ( E )$ of the event $E$ is defined as follows. $$I ( E ) = - \log _ { 2 } \mathrm { P } ( E )$$ Which of the following are correct? Select all that apply from . (Note: the probability that event $E$ occurs, $\mathrm { P } ( E )$, is positive, and the unit of information content is bits.) [4 points]
ㄱ. If event $E$ is rolling one die and getting an odd number, then $I ( E ) = 1$. ㄴ. If two events $A , B$ are independent and $\mathrm { P } ( A \cap B ) > 0$, then $I ( A \cap B ) = I ( A ) + I ( B )$. ㄷ. For two events $A , B$ with $\mathrm { P } ( A ) > 0 , \mathrm { P } ( B ) > 0$, we have $2 I ( A \cup B ) \leqq I ( A ) + I ( B )$.
(1) ㄱ
(2) ㄱ, ㄴ
(3) ㄱ, ㄷ
(4) ㄴ, ㄷ
(5) ㄱ, ㄴ, ㄷ

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) $\frac { 5 } { 6 }$