Question

If \[ 3P(A)=P(B)=\frac{3}{5} \] and \[ P(A\mid B)=\frac{2}{3}, \] then \[ P(A\cup B) \] is:

• A. \( \frac{3}{5} \)
• B. \( \frac{1}{5} \)
• C. \( \frac{2}{15} \)
• D. \( \frac{2}{5} \)

Hint:

Always decouple compound equalities like \( 3P(A) = P(B) = \text{value} \) into isolated algebraic equations first to prevent coefficient mistakes later.

Answer 1

The Correct Option is D

Concept: We use key equations from conditional probability and basic set operations:
• Multiplication rule: \( P(A \cap B) = P(B) \cdot P(A|B) \)
• Addition rule: \( P(A \cup B) = P(A) + P(B) - P(A \cap B) \)

Step 1: Extract individual probabilities. From the compound equality statement: \[ 3P(A) = \frac{3}{5} \implies P(A) = \frac{3}{5 \times 3} = \frac{1}{5} \] \[ P(B) = \frac{3}{5} \]

Step 2: Find the intersection probability \( P(A \cap B) \). Using the conditional probability values provided: \[ P(A|B) = \frac{P(A \cap B)}{P(B)} \implies P(A \cap B) = P(B) \cdot P(A|B) \] Substitute our values into this formula: \[ P(A \cap B) = \frac{3}{5} \times \frac{2}{3} = \frac{2}{5} \]

Step 3: Calculate the union probability \( P(A \cup B) \). Now apply the general probability addition rule: \[ P(A \cup B) = P(A) + P(B) - P(A \cap B) \] Substitute the calculated values into the formula: \[ P(A \cup B) = \frac{1}{5} + \frac{3}{5} - \frac{2}{5} = \frac{1 + 3 - 2}{5} = \frac{2}{5} \]