Question

(5 points) A multiple-choice examination has 15 questions, each with 5 possible answers only one of which is correct. Suppose that one of the students who takes the examination answers each of the questions with an independent random guess. What is the probability that the student answers at least four questions correctly? What is the expected number of questions that the student answers correctly? 4.

Answer 1

4)

Solution

Given that ,

p = 0.2

1 - p = 0.8

n = 15

Using binomial probability formula ,

P(X = x) = (_n C _x) * p^x * (1 - p)^{n - x}

P(X $\geq$ 4) = P(X < 4)

= P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3)

= (₁₅ C ₀) * 0.2⁰ * (0.8)15 +   (₁₅ C ₁) * 0.2¹ * (0.8)¹⁴ + (₁₅ C ₃) * 0.2³ * (0.8)¹³ + (₁₅ C 3) * 0.2³ * (0.8)¹²

= 0.3518

Probability = 0.3518

Expected number of questions = 15 * 0.2 = 3