Probability
Binomial Distribution
Grade 12
Question:
<p>A multiple choice examination has 5 questions. Each question has three alternative answers of which exactly one is correct. The probability that a student will get 4 or more correct answers just by guessing is</p>
<p>(a) \(\frac{17}{3^5}\)</p>
<p>(b) \(\frac{13}{3^5}\)</p>
<p>(c) \(\frac{11}{3^5}\)</p>
<p>(d) \(\frac{10}{3^5}\)</p>
Step-by-Step Solution
Key Concept: Use binomial distribution to calculate the probability of getting exactly 4 and exactly 5 correct answers, then sum them.
<p><strong>Step 1:</strong> Probability of success on each question = 1/3, probability of failure = 2/3</p><p><strong>Step 2:</strong> P(4 or more correct) = P(4 correct) + P(5 correct)</p><p><strong>Step 3:</strong> P(4 correct) = C(5,4) × (1/3)⁴ × (2/3)¹ = 5 × (1/81) × (2/3) = 10/243</p><p><strong>Step 4:</strong> P(5 correct) = C(5,5) × (1/3)⁵ × (2/3)⁰ = 1 × (1/243) × 1 = 1/243</p><p><strong>Step 5:</strong> P(4 or more) = 10/243 + 1/243 = 11/243 = 11/3⁵</p>
Correct Answer: C