<p>Three critics review a book. Odds in favour of the book are 5:2, 4:3 and 3:4 respectively for the three critics. Find the probability that majority are in favour of the book.</p>
Step-by-Step Solution
Key Concept: Convert odds to probabilities, then find the probability that at least 2 out of 3 critics favor the book by summing the relevant cases (exactly 2 favorable OR all 3 favorable).
<p><strong>Step 1: Convert odds to probabilities</strong></p><p>For critic 1: Odds 5:2 in favour → P₁ = 5/(5+2) = 5/7</p><p>For critic 2: Odds 4:3 in favour → P₂ = 4/(4+3) = 4/7</p><p>For critic 3: Odds 3:4 in favour → P₃ = 3/(3+4) = 3/7</p><p><strong>Step 2: Find complementary probabilities (against)</strong></p><p>Q₁ = 2/7, Q₂ = 3/7, Q₃ = 4/7</p><p><strong>Step 3: Majority means at least 2 critics in favour</strong></p><p>P(exactly 2 in favour) = P₁P₂Q₃ + P₁Q₂P₃ + Q₁P₂P₃</p><p>= (5/7)(4/7)(4/7) + (5/7)(3/7)(3/7) + (2/7)(4/7)(3/7)</p><p>= 80/343 + 45/343 + 24/343 = 149/343</p><p><strong>Step 4: Add probability of all 3 in favour</strong></p><p>P(all 3 in favour) = P₁P₂P₃ = (5/7)(4/7)(3/7) = 60/343</p><p><strong>Step 5: Total probability</strong></p><p>P(majority in favour) = 149/343 + 60/343 = <strong>209/343</strong></p>
Correct Answer: 209/343