EJU - Examination for Japanese University Mathematics: Probability and Sets Questions and Answers — Questions and Answers
Question 1: A bag contains 5 red balls and 3 white balls. If two balls are drawn from the bag at the same time, what is the probability that both balls are red?
- 3/14
- 5/14 (Correct answer)
- 5/8
- 25/64
Correct answer: 5/14
The total number of ways to choose 2 balls from 8 is given by the combination formula ₈C₂ = (8 * 7) / (2 * 1) = 28. The number of ways to choose 2 red balls from 5 is ₅C₂ = (5 * 4) / (2 * 1) = 10. Therefore, the probability is the ratio of favorable outcomes to total outcomes, which is 10/28, simplifying to 5/14.
Question 2: There are 6 students. In how many ways can a president, vice-president, and treasurer be chosen from this group, assuming no student can hold more than one office?
- 18
- 20
- 120 (Correct answer)
- 216
Correct answer: 120
This is a permutation problem because the order of selection matters (president, vice-president, and treasurer are distinct positions). The number of ways to arrange 3 students from a group of 6 is calculated as ₆P₃ = 6! / (6-3)! = 6 * 5 * 4 = 120.
Question 3: Let A and B be two sets such that n(A) = 15, n(B) = 20, and n(A ∩ B) = 5. What is the value of n(A ∪ B)?
- 25
- 30 (Correct answer)
- 35
- 40
Correct answer: 30
The number of elements in the union of two sets is given by the Principle of Inclusion-Exclusion: n(A ∪ B) = n(A) + n(B) - n(A ∩ B). Substituting the given values: n(A ∪ B) = 15 + 20 - 5 = 30.
Question 4: A single fair die is rolled. What is the probability that the number rolled is a prime number, given that the number is odd?
- 1/3
- 1/2
- 2/3 (Correct answer)
- 3/5
Correct answer: 2/3
The possible odd outcomes when rolling a die are {1, 3, 5}. This is our new sample space, with 3 possible outcomes. Among these, the prime numbers are {3, 5}. There are 2 favorable outcomes. Therefore, the conditional probability is 2/3. (Note: 1 is not a prime number).
Question 5: Let U = {x | x is an integer, 1 ≤ x ≤ 10} be the universal set. Let A = {1, 2, 3, 4} and B = {2, 4, 6, 8}. Which of the following represents the set (A ∪ B)ᶜ, the complement of the union of A and B?
- {1, 3, 5, 6, 7, 8, 9, 10}
- {5, 7, 9, 10} (Correct answer)
- {2, 4}
- {1, 3, 6, 8}
Correct answer: {5, 7, 9, 10}
First, find the union of A and B: A ∪ B = {1, 2, 3, 4, 6, 8}. The complement of this set, (A ∪ B)ᶜ, consists of all elements in the universal set U that are not in A ∪ B. The elements in U are {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}. Removing the elements of A ∪ B leaves {5, 7, 9, 10}.
Question 6: A coin is flipped 3 times. What is the probability of getting exactly two heads?
- 1/8
- 1/4
- 1/2
- 3/8 (Correct answer)
Correct answer: 3/8
The total number of possible outcomes is 2³ = 8 (HHH, HHT, HTH, THH, HTT, THT, TTH, TTT). The outcomes with exactly two heads are {HHT, HTH, THH}. There are 3 such favorable outcomes. Therefore, the probability is 3/8.
EJU - Examination for Japanese University Mathematics: Probability and Sets Questions and Answers
A bag contains 5 red balls and 3 white balls.
If two balls are drawn from the bag at the same time, what is the probability that both balls are red?