Question

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 __________

• A. 1 6
• B. 5 18
• C. 7 18
• D. 5 6

Hint:

For each value of the first roll \(a\), count how many of the six second-roll faces are multiples of \(a\); sum these counts and divide by 36 to get \(7/18\).

Answer 1

The Correct Option is C

Step 1: The dice is rolled twice, so the total number of equally likely outcomes is \(6 \times 6 = 36\), since each roll has 6 possible faces.

Step 2: We need the number of ordered pairs \((a, b)\), where \(a\) is the first roll and \(b\) is the second roll, such that \(b\) is an integer multiple of \(a\) (i.e., \(b = ka\) for some positive integer \(k\)), with \(a, b \in \{1,2,3,4,5,6\}\).

Step 3: Count favorable \(b\) values for each possible \(a\):
\(a=1\): multiples of 1 in range are 1,2,3,4,5,6 → 6 values.
\(a=2\): multiples of 2 in range are 2,4,6 → 3 values.
\(a=3\): multiples of 3 in range are 3,6 → 2 values.
\(a=4\): multiples of 4 in range are 4 → 1 value.
\(a=5\): multiples of 5 in range are 5 → 1 value.
\(a=6\): multiples of 6 in range are 6 → 1 value.

Step 4: Total favorable outcomes \(= 6+3+2+1+1+1 = 14\).

Step 5: Required probability \(= \dfrac{14}{36} = \dfrac{7}{18}\).

\[ P(\text{second is a multiple of first}) = \frac{7}{18} \]

Final Answer: (C) \(\dfrac{7}{18}\)