Question
After collecting data from a random sample, David finds that the probability a randomly selected person in his sample enjoys reading as a hobby is 0.46. He also finds that this probability is the same if he looks only at the high school students in his sample. Which of these describes his findings mathematically?
A.
The events "a person from his sample is a high school student" and "a person from his sample enjoys reading as a hobby" are mutually exclusive.
B.
The events "a person from his sample is a high school student" and "a person from his sample enjoys reading as a hobby" are dependent.
C.
The events "a person from his sample is a high school student" and "a person from his sample enjoys reading as a hobby" are independent.
D.
The events "a person from his sample is a high school student" and "a person from his sample enjoys reading as a hobby" are complementary.
A.
The events "a person from his sample is a high school student" and "a person from his sample enjoys reading as a hobby" are mutually exclusive.
B.
The events "a person from his sample is a high school student" and "a person from his sample enjoys reading as a hobby" are dependent.
C.
The events "a person from his sample is a high school student" and "a person from his sample enjoys reading as a hobby" are independent.
D.
The events "a person from his sample is a high school student" and "a person from his sample enjoys reading as a hobby" are complementary.
Answers
GPT-4o mini
To analyze David's findings, we need to consider the definitions of independent and dependent events in probability:
- Two events are **independent** if the occurrence of one does not affect the probability of the other. Mathematically, this means \(P(A \cap B) = P(A) \cdot P(B)\).
- Two events are **dependent** if the occurrence of one event affects the probability of the other.
In this scenario, David finds that the probability of enjoying reading as a hobby is 0.46 for the entire sample and remains 0.46 when looking exclusively at high school students. This indicates that knowing a person is a high school student does not change the probability of that person enjoying reading.
Thus, we can conclude that:
- The probability of enjoying reading remains the same regardless of being a high school student or not, indicating that the two events are independent.
Therefore, the correct answer that describes his findings mathematically is:
**C. The events "a person from his sample is a high school student" and "a person from his sample enjoys reading as a hobby" are independent.**
- Two events are **independent** if the occurrence of one does not affect the probability of the other. Mathematically, this means \(P(A \cap B) = P(A) \cdot P(B)\).
- Two events are **dependent** if the occurrence of one event affects the probability of the other.
In this scenario, David finds that the probability of enjoying reading as a hobby is 0.46 for the entire sample and remains 0.46 when looking exclusively at high school students. This indicates that knowing a person is a high school student does not change the probability of that person enjoying reading.
Thus, we can conclude that:
- The probability of enjoying reading remains the same regardless of being a high school student or not, indicating that the two events are independent.
Therefore, the correct answer that describes his findings mathematically is:
**C. The events "a person from his sample is a high school student" and "a person from his sample enjoys reading as a hobby" are independent.**