Probability
Conditional Probability
Grade 12
Question:
<p>Let \(X\) and \(Y\) be two events such that \(P(X) = \dfrac{1}{3}\), \(P(X|Y) = \dfrac{1}{2}\) and \(P(Y|X) = \dfrac{2}{5}\). Then</p>
<p>\(P(Y) = \dfrac{4}{15}\)</p>
<p>\(P(X'|Y) = \dfrac{1}{2}\)</p>
<p>\(P(X \cup Y) = \dfrac{2}{5}\)</p>
<p>\(P(X \cap Y) = \dfrac{1}{5}\)</p>
Step-by-Step Solution
Key Concept: Use the conditional probability formula P(A|B) = P(A∩B)/P(B) to find P(X∩Y), P(Y), and then derive P(X∪Y) and independence relationships. Verify each statement systematically using these foundational relationships.
<p><strong>Step 1: Find P(X∩Y)</strong></p><p>From P(Y|X) = P(X∩Y)/P(X):</p><p>2/5 = P(X∩Y)/(1/3)</p><p>P(X∩Y) = 2/5 × 1/3 = <strong>2/15</strong></p><p><strong>Step 2: Find P(Y)</strong></p><p>From P(X|Y) = P(X∩Y)/P(Y):</p><p>1/2 = (2/15)/P(Y)</p><p>P(Y) = (2/15) × 2 = <strong>4/15</strong></p><p><strong>Step 3: Check independence</strong></p><p>For independence: P(X∩Y) should equal P(X)·P(Y)</p><p>P(X)·P(Y) = 1/3 × 4/15 = 4/45 ≠ 2/15 = 6/45</p><p>So X and Y are <strong>NOT independent</strong></p><p><strong>Step 4: Find P(X∪Y)</strong></p><p>P(X∪Y) = P(X) + P(Y) - P(X∩Y)</p><p>= 1/3 + 4/15 - 2/15 = 5/15 + 4/15 - 2/15 = <strong>7/15</strong></p><p><strong>Step 5: Verify each option:</strong></p><p>A) P(Y) = 4/15 ✓</p><p>B) P(X∪Y) = 7/15 ✓</p><p>C) X and Y are independent ✗ (they are dependent)</p><p>D) P(X∩Y) = 2/15 ✓</p><p>∴ Answer: <strong>A, B, D</strong></p>
Correct Answer: A,B,D