40. An unprepared student makes random guess for ten MCQs question having 4 options, on a quiz. The probability of at most two correct answer is: A) \( \mathbf{0 . 5 2 5 5} \) B) 0.4744 C) 0.7251 D) 0.9437

To determine the probability that an unprepared student, who makes random guesses on a 10-question multiple-choice quiz (each question having 4 options), gets **at most two** correct answers, we can use the **binomial probability formula**: \[ P(X = k) = \binom{n}{k} \left(\frac{1}{4}\right)^k \left(\frac{3}{4}\right)^{n-k} \] Where: - \( n = 10 \) (total number of questions) - \( k \) = number of correct answers (0, 1, or 2) - \( p = \frac{1}{4} \) (probability of guessing a question correctly) We need to calculate \( P(X \leq 2) = P(X=0) + P(X=1) + P(X=2) \). ### Step-by-Step Calculation: 1. **Probability of 0 Correct Answers (\( P(X=0) \))**: \[ P(X=0) = \binom{10}{0} \left(\frac{1}{4}\right)^0 \left(\frac{3}{4}\right)^{10} = 1 \times 1 \times \left(\frac{3}{4}\right)^{10} \approx 0.0563 \] 2. **Probability of 1 Correct Answer (\( P(X=1) \))**: \[ P(X=1) = \binom{10}{1} \left(\frac{1}{4}\right)^1 \left(\frac{3}{4}\right)^9 = 10 \times 0.25 \times 0.0751 \approx 0.1877 \] 3. **Probability of 2 Correct Answers (\( P(X=2) \))**: \[ P(X=2) = \binom{10}{2} \left(\frac{1}{4}\right)^2 \left(\frac{3}{4}\right)^8 = 45 \times 0.0625 \times 0.1001 \approx 0.2816 \] 4. **Total Probability (\( P(X \leq 2) \))**: \[ P(X \leq 2) = P(X=0) + P(X=1) + P(X=2) \approx 0.0563 + 0.1877 + 0.2816 = 0.5256 \] ### Conclusion: The calculated probability is approximately **0.5256**, which closely matches option **A) 0.5255**. **Answer:** **A) 0.5255**