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Official PYQ · 2025Set 1 · Q8MCQ2 marks

GATE CS 2025 Set 1 Q8 — General Aptitude

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A fair six-faced dice, with the faces labelled  ‘ 11 ’, ‘ 22 ’, ‘ 33 ’, ‘ 44 ’, ‘ 55 ’, and ‘ 66 ’, is rolled thrice. What is the probability of rolling ‘ 66 ’ exactly once?

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Answer: A

We can model this problem using the binomial probability distribution. Let a "success" be rolling a '6' and a "failure" be rolling any other number.

For a single roll of a fair six-faced die:

  • Probability of success (rolling a '6'): p=16p = \dfrac{1}{6}
  • Probability of failure (not rolling a '6'): q=1−p=56q = 1 - p = \dfrac{5}{6}

The die is rolled n=3n = 3 times. We want to find the probability of getting exactly k=1k = 1 success.

Using the binomial probability formula:

P(X=k)=(nk)pkqn−kP(X = k) = \binom{n}{k} p^k q^{n-k}

Substitute the values into the formula:

P(X=1)=(31)(16)1(56)3−1P(X = 1) = \binom{3}{1} \left(\frac{1}{6}\right)^1 \left(\frac{5}{6}\right)^{3-1} P(X=1)=3×16×(56)2P(X = 1) = 3 \times \frac{1}{6} \times \left(\frac{5}{6}\right)^2 P(X=1)=3×16×2536=75216P(X = 1) = 3 \times \frac{1}{6} \times \frac{25}{36} = \frac{75}{216}

Therefore, the probability of rolling '6' exactly once is 75216\dfrac{75}{216}, which corresponds to option A.

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