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>\(\dfrac{17}{3^5}\)</p>
<p>\(\dfrac{13}{3^5}\)</p>
<p>\(\dfrac{11}{3^5}\)</p>
<p>\(\dfrac{10}{3^5}\)</p>
Step-by-Step Solution
Key Concept: Use binomial probability with n=5, p=1/3 (probability of correct guess), and sum P(X=4) + P(X=5) for getting 4 or more correct answers.
<p><strong>Step 1:</strong> Identify the binomial parameters: n = 5 questions, p = 1/3 (probability of guessing correctly), q = 2/3 (probability of guessing incorrectly).</p><p><strong>Step 2:</strong> Calculate P(X = 4) using the binomial formula: P(X = k) = C(n,k)p^k q^(n-k)</p><p>P(X = 4) = C(5,4)(1/3)^4(2/3)^1 = 5 × (1/81) × (2/3) = 10/243</p><p><strong>Step 3:</strong> Calculate P(X = 5):</p><p>P(X = 5) = C(5,5)(1/3)^5(2/3)^0 = 1 × (1/243) × 1 = 1/243</p><p><strong>Step 4:</strong> Find P(X ≥ 4) = P(X = 4) + P(X = 5):</p><p>P(X ≥ 4) = 10/243 + 1/243 = 11/243</p><p>∴ Answer: C</p>
Correct Answer: C