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) $\frac{7}{15}$

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) $\dfrac{46}{90}$

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.

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) $\frac { 3 } { 4 }$

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) $\frac { 5 } { 6 }$

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) $\frac { 3 } { 4 }$

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) $\frac { 1 } { 5 }$

The total number of students at a certain school is 360, and each student chose either experiential learning A or experiential learning B. Among the students at this school, those who chose experiential learning A are 90 male students and 70 female students. When one student is randomly selected from the students at this school who chose experiential learning B, the probability that this student is male is $\frac { 2 } { 5 }$. What is the number of female students at this school? [3 points]
(1) 180
(2) 185
(3) 190
(4) 195
(5) 200

There are two bags A and B, each containing 4 cards with the numbers $1,2,3,4$ written on them. Person 甲 draws two cards from bag A, and person 乙 draws two cards from bag B, each randomly. The probability that the sum of the numbers on the two cards held by 甲 equals the sum of the numbers on the two cards held by 乙 is $\frac { q } { p }$. Find the value of $p + q$. (Here, $p , q$ are coprime natural numbers.) [4 points]

A die is rolled three times, and the results are $a$, $b$, and $c$ in order. What is the probability that $a \times b \times c = 4$? [3 points]
(1) $\frac { 1 } { 54 }$
(2) $\frac { 1 } { 36 }$
(3) $\frac { 1 } { 27 }$
(4) $\frac { 5 } { 108 }$
(5) $\frac { 1 } { 18 }$

There are 5 cards with letters $\mathrm { A } , \mathrm { B } , \mathrm { C } , \mathrm { D } , \mathrm { E }$ written on them and 4 cards with numbers $1,2,3,4$ written on them. When all 9 cards are arranged in a line in random order using each card once, what is the probability that the card with letter A has number cards on both sides? [3 points]
(1) $\frac { 5 } { 12 }$
(2) $\frac { 1 } { 3 }$
(3) $\frac { 1 } { 4 }$
(4) $\frac { 1 } { 6 }$
(5) $\frac { 1 } { 12 }$

A supermarket randomly selected 1000 customers and recorded their purchasing of four products: A, B, C, and D. The data is organized in the table below, where ``✓'' indicates purchase and ``×'' indicates no purchase.\n\n\n

\n\n\backslashbox{Number of Customers}{Product}	A	B	C	D
\n\n100	✓	×	✓	✓
\n\n217	×	✓	×	✓
\n\n200	✓	✓	✓	×
\n\n300	✓	×	✓	×
\n\n85	✓	×	×	×
\n\n98	×	✓	×	×
\n\n

\n\n\n(I) Estimate the probability that a customer purchases both B and C\n(II) Estimate the probability that a customer purchases exactly 3 of the four products A, B, C, and D\n(III) If a customer has purchased product A, which of products B, C, and D is the customer most likely to have purchased?

16. (This question is worth 12 points)
A shopping mall is holding a promotional lottery activity. After customers purchase goods of a certain amount, they can participate in the lottery. The lottery method is as follows: randomly draw 1 ball each from box A containing 2 red balls $\mathrm { A } _ { 1 } , \mathrm { A } _ { 2 }$ and 1 white ball B, and from box B containing 2 red balls $\mathrm { a } _ { 1 } , \mathrm { a } _ { 2 }$ and 2 white balls $\mathrm { b } _ { 1 } , \mathrm { b } _ { 2 }$. If both balls drawn are red, the customer wins; otherwise, the customer does not win. (I) List all possible outcomes of drawing balls using the ball labels. (II) Someone claims: Since there are more red balls than white balls in both boxes, the probability of winning is greater than the probability of not winning. Do you agree? Please explain your reasoning.

5. A bag contains 4 balls of identical shape and size, including 1 white ball, 1 red ball, and 2 yellow balls. If 2 balls are randomly drawn at once, then the probability that the 2 balls have different colors is $\_\_\_\_$ .

Chinese mathematician Chen Jingrun achieved world-leading results in research on Goldbach's conjecture. Goldbach's conjecture states that ``every even number greater than 2 can be expressed as the sum of two prime numbers'', such as $30 = 7 + 23$. Among prime numbers not exceeding 30, if two different numbers are randomly selected, the probability that their sum equals 30 is
A. $\frac { 1 } { 12 }$
B. $\frac { 1 } { 14 }$
C. $\frac { 1 } { 15 }$
D. $\frac { 1 } { 18 }$