Question
After collecting data from a random sample, Ni’Lia finds that the probability a randomly selected person in the sample enjoys gaming as a hobby is .38. She also finds that the probability is the same if she looks only at the middle school students in the sample. Describe her findings mathematically.
Hint: what is the test for independence?
P(A intersect B) = P(A) * P(B).
Group of answer choices
The events are independent.
The events are dependent.
Hint: what is the test for independence?
P(A intersect B) = P(A) * P(B).
Group of answer choices
The events are independent.
The events are dependent.
Answers
GPT-4o mini
To describe Ni’Lia's findings mathematically, we can define two events:
- Let **A** be the event that a randomly selected person enjoys gaming as a hobby.
- Let **B** be the event that a randomly selected person is a middle school student.
From her findings, we know:
- \( P(A) = 0.38 \): The probability that a randomly selected person enjoys gaming.
- \( P(A|B) = 0.38 \): The probability that a middle school student enjoys gaming is also 0.38.
To determine if the two events, A and B, are independent, we check the definition of independence:
Two events A and B are independent if:
\[ P(A \cap B) = P(A) \cdot P(B) \]
We can also express independence in terms of conditional probability:
If \( P(A|B) = P(A) \), then A and B are independent.
Given that:
- \( P(A|B) = 0.38 \) (the probability that a middle school student enjoys gaming)
- \( P(A) = 0.38 \)
We see that \( P(A|B) = P(A) \).
Thus, since \( P(A|B) = P(A) \), we conclude that the events A and B are independent.
Therefore, the correct choice is:
**The events are independent.**
- Let **A** be the event that a randomly selected person enjoys gaming as a hobby.
- Let **B** be the event that a randomly selected person is a middle school student.
From her findings, we know:
- \( P(A) = 0.38 \): The probability that a randomly selected person enjoys gaming.
- \( P(A|B) = 0.38 \): The probability that a middle school student enjoys gaming is also 0.38.
To determine if the two events, A and B, are independent, we check the definition of independence:
Two events A and B are independent if:
\[ P(A \cap B) = P(A) \cdot P(B) \]
We can also express independence in terms of conditional probability:
If \( P(A|B) = P(A) \), then A and B are independent.
Given that:
- \( P(A|B) = 0.38 \) (the probability that a middle school student enjoys gaming)
- \( P(A) = 0.38 \)
We see that \( P(A|B) = P(A) \).
Thus, since \( P(A|B) = P(A) \), we conclude that the events A and B are independent.
Therefore, the correct choice is:
**The events are independent.**