Question

A student answers all 48 questions on a multiple-choice test by guessing. Each question has four possible answers, only one of which is correct. Find the probability that the student gets exactly 15 correct answers. Use the normal distribution to approximate the binomial distribution.

Answer 1

Let X be the answers all 48 questions on a multiple-choice test by guessing.

p = 1/4

q = 1 - 1/4 = 3/4

n = 48

μx = n*p = 48*(1/4) = 12

σx = sqrt(n*p*q) = sqrt(4/*(1/4)*(3/4)) = 0.8660254038

P(X = 15) = P(X ≤ 16) - P(X ≤ 14) = P((X - 12)/0.8660254038 ≤ (16 - 12)/0.8660254038) - P((X - 12)/0.8660254038 ≤ (14 - 12)/0.8660254038) = P(Z ≤ 4.62) - P(Z ≤ 2.31) = 1 - 0.9896 = 0.0104

You could use the binomial distribution:

P(X = 15) = 0.07670882173

(48)
(15)*((1/4)^15)*(3/4)^33 = 0.07670882173