To determine whether two events \( A \) and \( B \) are independent, you use the definition of independence in probability theory. Specifically, events \( A \) and \( B \) are independent if and only if:
\[ P(A \cap B) = P(A) \cdot P(B) \]
where:
- \( P(A) \) is the probability of event \( A \) occurring,
- \( P(B) \) is the probability of event \( B \) occurring,
- \( P(A \cap B) \) is the probability of both events \( A \) and \( B \) occurring simultaneously.
To test for independence, follow these steps:
- Calculate \( P(A) \) and \( P(B) \).
- Calculate \( P(A \cap B) \).
- Compare \( P(A \cap B) \) with \( P(A) \cdot P(B) \).
If \( P(A \cap B) = P(A) \cdot P(B) \), then events \( A \) and \( B \) are independent. If not, they are dependent.