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: use the test for independence? P(A intersect B) = P(A) * P(B).



Group of answer choices

The events are independent.

The events are dependent.

Answer 1

To describe Ni'Lia's findings mathematically, we need to define the events involved based on her sample:

- Let \( A \) represent the event that a randomly selected person enjoys gaming as a hobby.
- Let \( B \) represent the event that a randomly selected person is a middle school student.

From the information provided, we have:
- \( P(A) = 0.38 \) (the probability that a randomly selected person enjoys gaming)
- \( P(A|B) = 0.38 \) (the probability that a middle school student enjoys gaming)

To determine whether the events \( A \) and \( B \) are independent or dependent, we can use the definition of independence for events. Events \( A \) and \( B \) are independent if:

\[
P(A \cap B) = P(A) \times P(B)
\]

However, since we are not provided with \( P(B) \) or \( P(A \cap B) \), we can instead use an alternate condition for independence considering conditional probability:

Events \( A \) and \( B \) are independent if:

\[
P(A | B) = P(A)
\]

In this case, we find that:

- \( P(A | B) = 0.38 \)
- \( P(A) = 0.38 \)

Since \( P(A | B) = P(A) \), we conclude that the events \( A \) and \( B \) are independent.

### Conclusion:

The events are independent.