GATE CS 2025 Set 1 Q.8: Quantitative Aptitude

A fair six-faced dice, with the faces labelled '1', '2', '3', '4', '5', and '6', is rolled thrice. What is the probability of rolling '6' exactly once?

$\frac{75}{216}$

$\frac{1}{6}$

$\frac{1}{18}$

$\frac{25}{216}$

Official answer and solution

Official answer: A
From: Official GATE 2025 answer key

Solution

Model the three rolls as $n = 3$ independent Bernoulli trials with success probability $p = 1/6$ and failure probability $q = 5/6$ .

In each roll of a fair six-faced die, $P(\text{rolling '6'}) = \frac{1}{6}$ and $P(\text{not rolling '6'}) = \frac{5}{6}$ . Across $3$ independent rolls, the probability of rolling '6' exactly once is:

\binom{3}{1} \left(\frac{1}{6}\right)^1 \left(\frac{5}{6}\right)^2 = 3 \times \frac{1}{6} \times \frac{25}{36} = \frac{75}{216}

Final answer

Option (A)

Common trap

Forgetting to multiply by $\binom{3}{1} = 3$ for the three possible rolls on which the single '6' can occur (which gives $25/216$ ).

The answer is the one in the official GATE answer key. The solution is this site's own explanation.

Source

Paper: GATE 2025 Computer Science and Information Technology - Session 1 (CS1)
Question: 8 of 65
Section: General Aptitude

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