Probability
Independent Events
Grade 12
Question:
<p>\(A\) and \(B\) are two independent events. The probability that both \(A\) and \(B\) occur is \(\frac{1}{6}\) and the probability that neither of them occur is \(\frac{1}{3}\). Then \(P(A)\) is equal to</p>
<p>(a) \(\frac{1}{2}\)</p>
<p>(b) \(\frac{1}{3}\)</p>
<p>(c) \(\frac{5}{6}\)</p>
<p>(d) \(\frac{1}{2}\) or \(\frac{1}{3}\)</p>
Step-by-Step Solution
Key Concept: Use the independence property: P(A∩B) = P(A)·P(B) and P(A'∩B') = P(A')·P(B'). Set up two equations from the given probabilities to solve for P(A).
<p><strong>Step 1:</strong> Let P(A) = p and P(B) = q. Since A and B are independent:</p><p>P(A∩B) = P(A)·P(B) = pq = 1/6 ... (equation 1)</p><p><strong>Step 2:</strong> The probability that neither A nor B occurs:</p><p>P(A'∩B') = P(A')·P(B') = (1-p)(1-q) = 1/3 ... (equation 2)</p><p><strong>Step 3:</strong> Expand equation 2:</p><p>1 - p - q + pq = 1/3</p><p>1 - p - q + 1/6 = 1/3 (substituting pq = 1/6)</p><p>- p - q = 1/3 - 1 - 1/6 = 1/3 - 7/6 = -5/6</p><p>p + q = 5/6 ... (equation 3)</p><p><strong>Step 4:</strong> From equations 1 and 3, p and q are roots of:</p><p>t² - (5/6)t + 1/6 = 0</p><p>6t² - 5t + 1 = 0</p><p>(3t - 1)(2t - 1) = 0</p><p>t = 1/3 or t = 1/2</p><p><strong>Step 5:</strong> Therefore P(A) = 1/2 or 1/3. The answer typically designated is <strong>P(A) = 1/2</strong> (or 1/3 depending on convention, but standard form gives 1/2).</p><p>∴ Answer: D</p>
Correct Answer: D