Probability
Independent Events
Grade 12
Question:
<p>If \(A\) and \(B\) are two independent events, the probability that both \(A\) and \(B\) occur is \(\frac{1}{8}\) and the probability that neither of them occurs is \(\frac{3}{8}\). Find the probability of the occurrence of \(A\).</p>
<p>\(\frac{1}{2}\)</p>
<p>\(\frac{1}{4}\)</p>
<p>\(\frac{3}{4}\)</p>
<p>\(\frac{1}{2}\) or \(\frac{1}{4}\)</p>
Step-by-Step Solution
Key Concept: For independent events, use P(A∩B) = P(A)·P(B) and P(A'∩B') = P(A')·P(B') to set up equations, then solve the resulting quadratic where P(A) and P(B) are roots.
<p><strong>Step 1:</strong> Set up equations from given conditions.</p><p>For independent events A and B:</p><p>• P(A∩B) = P(A)·P(B) = 1/8</p><p>• P(A'∩B') = P(A')·P(B') = 3/8</p><p>Let P(A) = p and P(B) = q</p><p><strong>Step 2:</strong> Express the second condition.</p><p>P(A'∩B') = (1-p)(1-q) = 3/8</p><p>Expanding: 1 - p - q + pq = 3/8</p><p>Since pq = 1/8: 1 - p - q + 1/8 = 3/8</p><p>∴ p + q = 1 - 3/8 + 1/8 = 3/4</p><p><strong>Step 3:</strong> Solve for p using the system.</p><p>We have: pq = 1/8 and p + q = 3/4</p><p>So p and q are roots of: t² - (3/4)t + 1/8 = 0</p><p>Multiplying by 8: 8t² - 6t + 1 = 0</p><p>Using quadratic formula: t = (6 ± √(36-32))/16 = (6 ± 2)/16</p><p>∴ t = 1/2 or t = 1/4</p><p><strong>Step 4:</strong> Identify P(A).</p><p>Therefore P(A) = 1/2 or 1/4. The question asks for probability of occurrence of A, which typically refers to the larger value.</p><p>∴ <strong>Answer: P(A) = 1/2</strong> (or 1/4, depending on convention—usually 1/2 is selected as the primary answer)</p>
Correct Answer: D