Multiple Choice

1. An container contains $5$ red balls and $7$ blue balls. If $3$ balls are drawn at random without replacement, what is the probability that all $3$ balls are red?

   $5/12$

   $1/22$

   $3/10$

   $1/11$

2. A standard deck of $52$ playing cards is shuffled. If $2$ cards are drawn at random without replacement, what is the probability that both cards are kings?

   $4/52$

   $1/26$

   $1/169$

   $1/221$

3. A committee of $4$ people is to be chosen from a group of $7$ men and $6$ women. What is the probability that the committee consists of exactly $2$ men and $2$ women?

   $35/143$

   $7/13$

   $63/143$

   $1/2$

4. What is the probability that the letters of the word MATH are arranged in alphabetical order (A, H, M, T)?

   $1/4$

   $1/6$

   $1/24$

   $1/12$

5. Five friends (Alice, Bob, Carol, David, Emily) are sitting in a row. What is the probability that Alice and Bob sit next to each other?

   $1/2$

   $3/5$

   $1/5$

   $2/5$

6. A bag contains $4$ red marbles and $5$ blue marbles. If $3$ marbles are drawn at random, what is the probability that at least one marble is red?

   $37/42$

   $5/42$

   $4/9$

   $1/9$

7. From a box containing $10$ items, $3$ of which are defective, $4$ items are chosen at random. What is the probability that exactly $1$ of the chosen items is defective?

   $3/10$

   $1/2$

   $1/4$

   $7/10$

8. In a mini-lottery, $3$ numbers are chosen from $10$ distinct numbers. What is the probability of matching exactly $2$ of the $3$ winning numbers?

   $7/10$

   $7/40$

   $1/120$

   $3/10$

9. A box contains $6$ red, $4$ blue, and $2$ green balls. If $3$ balls are drawn at random, what is the probability that they are all of different colors?

   $1/220$

   $12/55$

   $1/11$

   $6/55$

10. Three students (X, Y, Z) are randomly assigned to three different classrooms (1, 2, 3), one student per classroom. What is the probability that none of them are assigned to their preferred classroom (X prefers 1, Y prefers 2, Z prefers 3)?

    $1/6$

    $1/2$

    $2/3$

    $1/3$

11. What is the probability that a randomly formed $4$-digit number using the digits $1, 2, 3, 4$ (without repetition) is an even number?

    $1/4$

    $1/3$

    $2/3$

    $1/2$

12. From a standard deck of $52$ cards, $5$ cards are drawn without replacement. Which of the following expressions represents the probability that at least one of them is a heart?

    $C(13,5)/C(52,5)$

    $1 - C(39,5)/C(52,5)$

    $C(13,1)/C(52,5)$

    $1 - C(13,5)/C(52,5)$

13. There are $8$ books on a shelf, including a science book and a math book. If the books are arranged randomly, what is the probability that the science book and math book are next to each other?

    $1/4$

    $1/2$

    $1/56$

    $1/8$

14. A group of $3$ friends are randomly assigned a day of the week for an appointment (e.g., Monday, Tuesday, etc.). What is the probability that all $3$ friends are assigned different days of the week?

    $30/49$

    $6/49$

    $1/7$

    $1/343$

15. A fair coin is tossed $5$ times. What is the probability of getting exactly $3$ heads?

    $3/5$

    $1/32$

    $5/16$

    $1/2$

16. From a set of $5$ distinct positive integers $\{1, 2, 3, 4, 5\}$, $2$ numbers are chosen at random without replacement. What is the probability that their sum is even?

    $1/5$

    $1/2$

    $3/5$

    $2/5$

17. $4$ people are to be seated randomly around a circular table. What is the probability that two specific people, A and B, sit next to each other?

    $1/3$

    $2/3$

    $1/4$

    $1/2$

18. What is the probability that a random arrangement of the letters in the word ``MISSISSIPPI'' begins and ends with `S'?

    $1/55$

    $6/55$

    $1/11$

    $12/55$

19. A box contains $3$ red, $4$ blue, and $5$ green balls. If $2$ balls are drawn without replacement, what is the probability that they are of the same color?

    $1/3$

    $1/6$

    $1/2$

    $19/66$

20. A class has $15$ boys and $10$ girls. If $3$ students are selected randomly for a presentation, what is the probability that at least one boy and at least one girl are selected?

    $1/4$

    $1/2$

    $69/92$

    $23/92$