A test has 20 multiple-choice questions, each with 4 options of which only one is correct. A student answers every question by pure guessing. The variance of the number of correct answers is
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → 3.75
Let X be the number of correct answers. Each guess is correct with probability \(\frac{1}{4}\), independently, so X follows a binomial distribution with
\( \displaystyle n = 20,\quad p = \frac{1}{4},\quad q = \frac{3}{4} \)
The variance of a binomial distribution is npq:
\( \displaystyle \mathrm{Var}(X) = 20 \times \frac{1}{4} \times \frac{3}{4} \)
\( \displaystyle = \frac{60}{16} = 3.75 \)
Why the other options are incorrect
Option (1) 5 — Incorrect. 5 is the mean np; the variance is npq.
Option (3) 15 — Incorrect. 15 is nq, the expected number of wrong answers.
Option (4) 0.1875 — Incorrect. This is pq alone; it must be multiplied by n = 20.