Question

If A and B are two events and $ P(A \cup B) = \frac{5}{6} $, $ P(A \cap B) = \frac{1}{3} $, $ P(\overline{B}) = \frac{1}{2} $, then A and B are

• A. Dependent
• B. Independent
• C. Mutually Exclusive
• D. None of these

Hint:

Two events are independent if \( P(A \cap B) = P(A) \cdot P(B) \). If this is not true, the events are dependent.

Answer 1

The Correct Option is A

Step 1: Use the Formula for the Union of Events
We know that: \[ P(A \cup B) = P(A) + P(B) - P(A \cap B) \] We are given: \[ P(A \cup B) = \frac{5}{6}, \quad P(A \cap B) = \frac{1}{3} \] From the formula, we can solve for \( P(A) + P(B) \).
Step 2: Determine Dependence
If \( A \) and \( B \) were independent, we would have: \[ P(A \cap B) = P(A) \cdot P(B) \] However, from the given data, \( P(A \cap B) \) does not equal \( P(A) \cdot P(B) \), implying that \( A \) and \( B \) are dependent.
Step 3: Conclusion
Thus, the correct answer is that A and B are dependent events.