Question

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

• A. Independent
• B. Dependent
• C. Mutually exclusive
• D. Exclusive

Hint:

Events \( A \) and \( B \) are independent if \( P(A \cap B) = P(A) \cdot P(B) \). Use the union formula \( P(A \cup B) = P(A) + P(B) - P(A \cap B) \) to find missing probabilities.

Answer 1

The Correct Option is A

Step 1: Use the given probabilities to find \( P(A) \).
We are given:
\( P(A \cap B) = \frac{1}{3} \),
\( P(A \cup B) = \frac{5}{6} \),
\( P(B) = \frac{1}{2} \).
Use the formula for the union of two events: \[ P(A \cup B) = P(A) + P(B) - P(A \cap B). \] Substitute the given values: \[ \frac{5}{6} = P(A) + \frac{1}{2} - \frac{1}{3}. \] Solve for \( P(A) \): \[ \frac{5}{6} = P(A) + \frac{3}{6} - \frac{2}{6}, \] \[ \frac{5}{6} = P(A) + \frac{1}{6}, \] \[ P(A) = \frac{5}{6} - \frac{1}{6} = \frac{4}{6} = \frac{2}{3}. \] So, \( P(A) = \frac{2}{3} \). Step 2: Check for independence.
Two events \( A \) and \( B \) are independent if: \[ P(A \cap B) = P(A) \cdot P(B). \] Compute \( P(A) \cdot P(B) \): \[ P(A) \cdot P(B) = \frac{2}{3} \cdot \frac{1}{2} = \frac{2}{6} = \frac{1}{3}. \] The given \( P(A \cap B) = \frac{1}{3} \), which matches: \[ P(A \cap B) = P(A) \cdot P(B). \] Thus, \( A \) and \( B \) are independent. Step 3: Check other properties to confirm.
Mutually exclusive: Events are mutually exclusive if \( P(A \cap B) = 0 \). Here, \( P(A \cap B) = \frac{1}{3} \neq 0 \), so they are not mutually exclusive.
Dependent: Events are dependent if they are not independent. Since \( P(A \cap B) = P(A) \cdot P(B) \), they are not dependent.
Exclusive: This term is ambiguous but often means mutually exclusive, which we already ruled out.
Step 4: Evaluate the options.
(1) Independent: Correct, as \( P(A \cap B) = P(A) \cdot P(B) \). Correct.
(2) Dependent: Incorrect, as the events are independent. Incorrect.
(3) Mutually exclusive: Incorrect, as \( P(A \cap B) \neq 0 \). Incorrect.
(4) Exclusive: Incorrect, assuming it means mutually exclusive. Incorrect.
Step 5: Select the correct answer.
The events \( A \) and \( B \) are independent, matching option (1).