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. \(\dfrac{1}{6}\)
• B. \(\dfrac{5}{18}\)
• C. \(\dfrac{7}{18}\)
• D. \(\dfrac{5}{6}\)

Hint:

For each possible first roll value \(a\), count how many values from 1 to 6 are exact multiples of \(a\), sum these counts, and divide by 36.

Answer 1

The Correct Option is C

Step 1: Use a divisor counting formula instead of listing multiples by hand.
For a fixed first roll \(a\), the values of \(b\) in \(\{1,\dots,6\}\) that are multiples of \(a\) are exactly \(a, 2a, 3a, \dots\) up to 6, so the count of favourable \(b\) values equals \(\left\lfloor \dfrac{6}{a} \right\rfloor\), the integer part of 6 divided by \(a\).

Step 2: Evaluate the floor function for each \(a\) from 1 to 6.
\(a=1: \lfloor 6/1 \rfloor = 6\)
\(a=2: \lfloor 6/2 \rfloor = 3\)
\(a=3: \lfloor 6/3 \rfloor = 2\)
\(a=4: \lfloor 6/4 \rfloor = \lfloor 1.5 \rfloor = 1\)
\(a=5: \lfloor 6/5 \rfloor = \lfloor 1.2 \rfloor = 1\)
\(a=6: \lfloor 6/6 \rfloor = 1\)

Step 3: Sum the floor values.
\[ \sum_{a=1}^{6} \left\lfloor \frac{6}{a} \right\rfloor = 6+3+2+1+1+1 = 14 \]
This sum is exactly the count of ordered pairs \((a,b)\) with \(a\) dividing \(b\), since each term counts the multiples of \(a\) available in the range.

Step 4: Since each value of \(a\) occurs with probability \(\frac{1}{6}\) and, given \(a\), the conditional probability that \(b\) is a multiple of \(a\) is \(\dfrac{\lfloor 6/a \rfloor}{6}\), the total probability is the average of these conditional probabilities:
\[ P = \frac{1}{6}\sum_{a=1}^{6} \frac{\lfloor 6/a \rfloor}{6} = \frac{14}{36} = \frac{7}{18} \]

Step 5: Conclude.
The floor sum method reproduces the same count of 14 favourable outcomes out of 36 obtained by direct listing, confirming the probability.
\[ \boxed{\dfrac{7}{18}} \]