Probability
Independent Events
Grade 12
Question:
<p>Three critics review a book. Odds in favor of the book are 5:2, 4:3, and 3:4, respectively, for the three critics. The probability that majority are in favor of the book is</p>
<p>35/49</p>
<p>125/343</p>
<p>164/343</p>
<p>209/343</p>
Step-by-Step Solution
Key Concept: Convert odds to probabilities for each critic, then find all scenarios where at least 2 out of 3 critics are in favor using the complement or direct counting method.
<p><strong>Step 1:</strong> Convert odds to probabilities for each critic.</p><p>Critic 1: Odds 5:2 → P₁ = 5/(5+2) = 5/7, Q₁ = 2/7</p><p>Critic 2: Odds 4:3 → P₂ = 4/(4+3) = 4/7, Q₂ = 3/7</p><p>Critic 3: Odds 3:4 → P₃ = 3/(3+4) = 3/7, Q₃ = 4/7</p><p><strong>Step 2:</strong> Find probability that majority (at least 2) are in favor.</p><p>P(majority in favor) = P(exactly 2 in favor) + P(all 3 in favor)</p><p><strong>Step 3:</strong> Calculate exactly 2 in favor:</p><p>= 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:</strong> Calculate all 3 in favor:</p><p>= P₁P₂P₃ = (5/7)(4/7)(3/7) = 60/343</p><p><strong>Step 5:</strong> Add both cases:</p><p>P(majority) = 149/343 + 60/343 = <strong>209/343</strong></p><p>∴ Answer: C</p>
Correct Answer: C