Two sets $$A = \{ 3,5,7,9 \} , B = \{ 3,7 \}$$ For the sets above, when $A - B = \{ a , 9 \}$, what is the value of $a$? [2 points] (1) 1 (2) 2 (3) 3 (4) 4 (5) 5
A bag contains 7 marbles, each labeled with a natural number from 2 to 8. When 2 marbles are drawn simultaneously from the bag, what is the probability that the two natural numbers on the drawn marbles are coprime? [3 points] (1) $\frac { 8 } { 21 }$ (2) $\frac { 10 } { 21 }$ (3) $\frac { 4 } { 7 }$ (4) $\frac { 2 } { 3 }$ (5) $\frac { 16 } { 21 }$
Point A is at the origin of the coordinate plane. The following trial is performed using one coin. Flip the coin once. If heads appears, move point A by 1 in the positive direction of the $x$-axis; if tails appears, move point A by 1 in the positive direction of the $y$-axis. Repeat this trial until the $x$-coordinate or $y$-coordinate of point A becomes 3 for the first time, then stop. What is the probability that when the $y$-coordinate of point A becomes 3 for the first time, the $x$-coordinate of point A is 1? [4 points] (1) $\frac { 1 } { 4 }$ (2) $\frac { 5 } { 16 }$ (3) $\frac { 3 } { 8 }$ (4) $\frac { 7 } { 16 }$ (5) $\frac { 1 } { 2 }$
For two sets $A = \{ a + 2,6 \} , B = \{ 3 , b - 1 \}$, when $A = B$, what is the value of $a + b$? (Note: $a , b$ are real numbers.) [2 points] (1) 5 (2) 6 (3) 7 (4) 8 (5) 9
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) $\frac { 1 } { 18 }$
A bag contains 5 balls labeled with the numbers $3, 3, 4, 4, 4$, one each. Using this bag and one die, a trial is performed to obtain a score according to the following rule: If the ball drawn from the bag is labeled 3, roll the die 3 times and the sum of the three results is the score. If the ball drawn from the bag is labeled 4, roll the die 4 times and the sum of the four results is the score. What is the probability that the score obtained from one trial is 10 points? Express this as $\frac { q } { p }$. Find the value of $p + q$. (Here, $p$ and $q$ are coprime natural numbers.) [1 point]
A bag contains 10 cards with natural numbers from 1 to 10 written on them, one number per card. When drawing 3 cards simultaneously at random from the bag, what is the probability that the smallest of the three natural numbers on the drawn cards is at most 4 or at least 7? [3 points] (1) $\frac { 4 } { 5 }$ (2) $\frac { 5 } { 6 }$ (3) $\frac { 13 } { 15 }$ (4) $\frac { 9 } { 10 }$ (5) $\frac { 14 } { 15 }$
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) $\frac { 10 } { 13 }$
A bag contains 1 white ball marked with 1, 1 white ball marked with 2, 1 black ball marked with 1, and 3 black balls marked with 2. We perform a trial of simultaneously drawing 3 balls from the bag. Let $A$ be the event that among the 3 balls drawn, 1 is white and 2 are black, and let $B$ be the event that the product of the numbers on the 3 balls is 8. What is the value of $\mathrm { P } ( A \cup B )$? [3 points] (1) $\frac { 11 } { 20 }$ (2) $\frac { 3 } { 5 }$ (3) $\frac { 13 } { 20 }$ (4) $\frac { 7 } { 10 }$ (5) $\frac { 3 } { 4 }$
There are 6 cards with natural numbers 1 through 6 written on the front and 0 written on the back. These 6 cards are placed so that the natural number $k$ is visible in the $k$-th position for natural numbers $k$ not exceeding 6. Using these 6 cards and one die, we perform the following trial: Roll the die once. If the result is $k$, flip the card in the $k$-th position and place it back in its original position. After repeating this trial 3 times, given that the sum of all numbers visible on the 6 cards is even, what is the probability that the die shows 1 exactly once? The probability is $\frac { q } { p }$. Find the value of $p + q$. (Here, $p$ and $q$ are coprime natural numbers.) [4 points]
There are 6 cards with the numbers $1, 2, 3, 4, 5, 6$ written on them, one number per card. When all 6 cards are arranged in a row in random order using each card exactly once, find the probability that the sum of the two numbers on the cards at both ends is at most 10. [3 points] (1) $\frac{8}{15}$ (2) $\frac{19}{30}$ (3) $\frac{11}{15}$ (4) $\frac{5}{6}$ (5) $\frac{14}{15}$
For two events $A$ and $B$, $$\mathrm{P}(A \mid B) = \mathrm{P}(A) = \frac{1}{2}, \quad \mathrm{P}(A \cap B) = \frac{1}{5}$$ What is the value of $\mathrm{P}(A \cup B)$? [3 points] (1) $\frac{1}{2}$ (2) $\frac{3}{5}$ (3) $\frac{7}{10}$ (4) $\frac{4}{5}$ (5) $\frac{9}{10}$
Five coins are placed in a line on a table. At the start, the coins in the 1st and 2nd positions show heads, and the coins in the remaining 3 positions show tails. Using these 5 coins and one die, the following trial is performed. Roll the die once. If the result is $k$, if $k \leq 5$, flip the coin in the $k$-th position once and place it back, if $k = 6$, flip all coins once and place them back. After repeating this trial 3 times, what is the probability that all 5 coins show heads? Express the answer as $\frac{q}{p}$. What is the value of $p + q$? (Given: $p$ and $q$ are coprime natural numbers.) [4 points]
A bag contains 5 white balls with the numbers $1,2,3,4,5$ written on them one each, and 5 black balls with the numbers $2,3,4,5,6$ written on them one each. When 2 balls are drawn simultaneously at random from the bag, what is the probability that the 2 balls drawn are either the same color or have the same number written on them? [3 points] (1) $\frac { 7 } { 15 }$ (2) $\frac { 8 } { 15 }$ (3) $\frac { 3 } { 5 }$ (4) $\frac { 2 } { 3 }$ (5) $\frac { 11 } { 15 }$
9. A card is randomly drawn from a shuffled deck of playing cards (52 cards). Event $A$ is ``drawing the King of Hearts'', and event $B$ is ``drawing a Spade''. Then the probability $P ( A \cup B ) = $ $\_\_\_\_$ $\frac { 7 } { 26 }$ (express the result as a fraction in lowest terms). Analysis: This examines the probability formula for mutually exclusive events. $P ( A \cup B ) = \frac { 1 } { 52 } + \frac { 13 } { 52 } = \frac { 7 } { 26 }$
10. From a shuffled deck of playing cards (52 cards), 2 cards are randomly drawn. The probability that ``both cards drawn are hearts'' is $\_\_\_\_$ (express the result as a fraction in lowest terms).
13. Among 9 randomly selected students, the probability that at least 2 students are born in the same month is $\_\_\_\_$ (assuming each month has the same number of days; round the result to 0.001)
1. Given sets $A = \{ 1,2,3 \} , B = \{ 2,3 \}$, then A. $\mathrm { A } = \mathrm { B }$ B. $\mathrm { A } \cap \mathrm { B } = \varnothing$ C. $A \subset B$ D. $B \subset A$