An exam consists of 10 multiple‑choice questions. Each question has four options of which exactly one is correct. A student guesses each answer independently. Let $X$ be the number of correct answers.

hard 3 marks Class 12 · Mathematics · Probability

Answer

PMF:

P (X = k) = (k 10) (\frac{1}{4})^{k} (\frac{3}{4})^{10 - k}, k = 0, 1, \dots, 10

P (X = 4) = (4 10) (\frac{1}{4})^{4} (\frac{3}{4})^{6} = \frac{153090}{1048576} \approx 0.146

E [X] = 2.5

Var (X) = \frac{15}{8} = 1.875

Step-by-step solution

Recognise that each guess is a Bernoulli trial with success probability $p = 1/4$ and failure probability $q = 3/4$ .
For $n = 10$ independent trials, $X$ follows a binomial distribution: $P (X = k) = (k 10) p^{k} q^{10 - k}$ for $k = 0, 1, \dots, 10$ .
Substitute $p = 1/4$ , $q = 3/4$ to obtain the required PMF.
For $k = 4$ , evaluate $P (X = 4) = (4 10) (1/4)^{4} (3/4)^{6}$ . Compute $(4 10) = 210$ , $(1/4)^{4} = 1/256$ , $(3/4)^{6} = 729/4096$ . Multiply: $210 \times 1/256 \times 729/4096 = 153090/1048576 \approx 0.146$ .
The mean of a binomial $n p$ gives $E [X] = 10 \times \frac{1}{4} = 2.5$ .
The variance $n pq$ gives $Var (X) = 10 \times \frac{1}{4} \times \frac{3}{4} = \frac{30}{16} = \frac{15}{8} = 1.875$ .
State all results together as the final answer.

Common mistake: Using

p = 1/5

instead of

1/4

or forgetting the factor

q

in the variance formula.

Concept tested: Binomial distribution (mean & variance)