<p>Let \(E\) and \(F\) be two independent events. The probability that both \(E\) and \(F\) happen is \(1/12\) and the probability that neither \(E\) nor \(F\) happens is \(1/2\), then a value of \(P(E)/P(F)\) is</p>
Step-by-Step Solution
Key Concept: Use the independence condition P(E∩F) = P(E)·P(F) = 1/12 and the complement condition P(E'∩F') = P(E')·P(F') = 1/2 to set up a system of equations in P(E) and P(F).
<p><strong>Step 1:</strong> Set P(E) = p and P(F) = q where p, q ∈ [0,1].</p><p><strong>Step 2:</strong> From 'both E and F happen': P(E∩F) = pq = 1/12</p><p><strong>Step 3:</strong> From 'neither E nor F happens': P(E'∩F') = (1-p)(1-q) = 1/2</p><p><strong>Step 4:</strong> Expand the second equation: 1 - p - q + pq = 1/2</p><p><strong>Step 5:</strong> Substitute pq = 1/12: 1 - p - q + 1/12 = 1/2</p><p><strong>Step 6:</strong> Simplify: p + q = 1 + 1/12 - 1/2 = 7/12</p><p><strong>Step 7:</strong> Now p and q satisfy: p + q = 7/12 and pq = 1/12. These are roots of t² - (7/12)t + 1/12 = 0</p><p><strong>Step 8:</strong> Multiply by 12: 12t² - 7t + 1 = 0, which factors as (3t - 1)(4t - 1) = 0</p><p><strong>Step 9:</strong> So t = 1/3 or t = 1/4, meaning {p,q} = {1/3, 1/4}</p><p><strong>Step 10:</strong> Therefore P(E)/P(F) = (1/3)/(1/4) = 4/3 or (1/4)/(1/3) = 3/4</p><p>∴ Answer: D (one of these ratios, depending on the options given)</p>
Correct Answer: D