A multiple choice examination has 5 questions. Each question has three alternative answers out of which exactly one is correct. The probability that a student will get 4 or more correct answers just by guessing is :

We are told that the test has $$n = 5$$ questions and that each question offers three alternatives, of which exactly one is correct. Therefore for every single question, the probability of marking the correct alternative purely by guessing is

$$p = \frac{1}{3},$$

while the probability of marking a wrong alternative is

$$q = 1 - p = 1 - \frac{1}{3} = \frac{2}{3}.$$

Let the random variable $$X$$ denote “the number of correct answers obtained by the student.” Because the questions are independent and the probability of success on each question is the same, $$X$$ follows a binomial distribution. The formula for the probability of getting exactly $$r$$ successes in $$n$$ independent trials is

$$P(X = r) \;=\; {n \choose r}\, p^{\,r}\, q^{\,n-r},$$

where $${n \choose r} = \dfrac{n!}{r!\,(n-r)!}$$ is the binomial coefficient. In our problem, $$n = 5, \; p = \dfrac{1}{3}, \; q = \dfrac{2}{3}.$$

We are asked to find the probability that the student gets “4 or more correct answers,” that is, $$X \ge 4.$$ Hence we must compute

$$P(X \ge 4) = P(X = 4) + P(X = 5).$$

First we evaluate $$P(X = 4):$$

$$\begin{aligned} P(X = 4) &= {5 \choose 4}\, p^{\,4}\, q^{\,5-4} \\ &= 5 \left(\frac{1}{3}\right)^{4} \left(\frac{2}{3}\right)^{1} \\ &= 5 \cdot \frac{1}{3^{4}} \cdot \frac{2}{3} \\ &= 5 \cdot \frac{2}{3^{5}} \\ &= \frac{10}{3^{5}}. \end{aligned}$$

Next we evaluate $$P(X = 5):$$

$$\begin{aligned} P(X = 5) &= {5 \choose 5}\, p^{\,5}\, q^{\,5-5} \\ &= 1 \left(\frac{1}{3}\right)^{5} \left(\frac{2}{3}\right)^{0} \\ &= 1 \cdot \frac{1}{3^{5}} \cdot 1 \\ &= \frac{1}{3^{5}}. \end{aligned}$$

Now we add these two probabilities to obtain $$P(X \ge 4):$$

$$\begin{aligned} P(X \ge 4) &= P(X = 4) + P(X = 5) \\ &= \frac{10}{3^{5}} + \frac{1}{3^{5}} \\ &= \frac{11}{3^{5}}. \end{aligned}$$

Thus the probability that the student will mark at least four answers correctly purely by guessing is

$$\boxed{\dfrac{11}{3^{5}}}.$$

Hence, the correct answer is Option A.