GATE 2026 CS (CS1) – Question 10
An unbiased six-faced dice whose faces are marked with numbers 1, 2, 3, 4, 5, and 6 is rolled twice in succession and the number on the top face is recorded each time. The probability that the number appearing in the second roll is an integer multiple of the number appearing in the first roll is __________
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Correct answer: (C) $\frac{7}{18}$
Explanation
Let the outcome of the first roll be $x \in \{1, 2, 3, 4, 5, 6\}$ and the second roll be $y \in \{1, 2, 3, 4, 5, 6\}$.
The sample space has $|S| = 6 \times 6 = 36$ equally likely outcomes.
We need $y$ to be an integer multiple of $x$:
- If $x = 1$: $y \in \{1, 2, 3, 4, 5, 6\}$ (6 outcomes)
- If $x = 2$: $y \in \{2, 4, 6\}$ (3 outcomes)
- If $x = 3$: $y \in \{3, 6\}$ (2 outcomes)
- If $x = 4$: $y \in \{4\}$ (1 outcome)
- If $x = 5$: $y \in \{5\}$ (1 outcome)
- If $x = 6$: $y \in \{6\}$ (1 outcome)
Total number of favorable outcomes:
$$N = 6 + 3 + 2 + 1 + 1 + 1 = 14$$
Thus, the required probability is:
$$P = \frac{14}{36} = \frac{7}{18}$$
Therefore, option (C) is correct.