Topic: Engineering Mathematics
A test has 5 multiple-choice questions. Each question has 4 answer options (A, B, C, D). What is the probability that a student will choose "B" for at least three questions if he/she leaves no questions blank?
- A. 1/1024
- B. 1/64
- C. 53/512
- D. 29/128
Answer: C. 53/512
Explanation: Probability that a student will answer B is $p = 1/4$. Probability that a student will not answer B is $q = 3/4$. The probability of choosing B for at least three questions is given by the binomial distribution: $P(X \ge 3) = ^5C_3 p³ q² + ^5C_4 p^4 q^1 + ^5C_5 p^5 q^0 = ^5C_3(1/4)^3(3/4)^2 + ^5C_4(1/4)^4(3/4) + ^5C_5(1/4)^5 = \frac{10 \times 9}{1024} + \frac{5 \times 3}{1024} + \frac{1}{1024} = \frac{90+15+1}{1024} = \frac{106}{1024} = \frac{53}{512}$.