Probability
Independent Events
Grade 12
Question:
<p>Four persons can hit a target correctly with probabilities \(1/2, 1/3, 1/4\) and \(1/8\) respectively. If all hit at the target independently, then the probability that the target would be hit, is:</p>
<p>\(\dfrac{25}{192}\)</p>
<p>\(\dfrac{7}{32}\)</p>
<p>\(\dfrac{1}{192}\)</p>
<p>\(\dfrac{25}{32}\)</p>
Step-by-Step Solution
Key Concept: The target is hit if at least one person hits it. Use the complement: P(at least one hit) = 1 - P(no one hits). Calculate P(all miss) by multiplying individual miss probabilities.
<p><strong>Step 1:</strong> Identify the probabilities of hitting: P₁ = 1/2, P₂ = 1/3, P₃ = 1/4, P₄ = 1/8</p><p><strong>Step 2:</strong> Find probabilities of missing: q₁ = 1/2, q₂ = 2/3, q₃ = 3/4, q₄ = 7/8</p><p><strong>Step 3:</strong> Since all four shoot independently, probability that none hit (all miss):</p><p>P(all miss) = (1/2) × (2/3) × (3/4) × (7/8) = 42/384 = 7/64</p><p><strong>Step 4:</strong> Probability that target is hit (at least one hits):</p><p>P(at least one hit) = 1 - P(all miss) = 1 - 7/64 = 57/64</p><p>∴ Answer: <strong>57/64</strong></p>
Correct Answer: D