An unbiased six faced dice whose faces are marked with 1, 2, 3, 4, 5 and 6 is rolled twice. The probability that the number appearing in the second roll is an integer multiple of the number appearing in the first roll?

The correct option is 7/18.

Let us break down the solution step-by-step to understand why this is the correct answer.

Step 1: Determine the total number of possible outcomes
When a fair six-faced die is rolled twice, each roll has 6 possible outcomes (1, 2, 3, 4, 5, or 6).
Therefore, the total number of outcomes in the sample space, denoted as N, is given by:
$$N=6\times 6=36$$

Step 2: Identify the favorable outcomes
Let the outcome of the first roll be x and the outcome of the second roll be y. We are looking for pairs (x, y) where y is an integer multiple of x. This means that y = k × x for some positive integer k, where both x and y belong to the set {1, 2, 3, 4, 5, 6}.
Let us analyze the possibilities for each value of the first roll x:

1. If the first roll x = 1, the second roll y can be any multiple of 1. All six outcomes are multiples of 1:
Favorable pairs: (1, 1), (1, 2), (1, 3), (1, 4), (1, 5), (1, 6) → 6 outcomes.

2. If the first roll x = 2, the second roll y must be a multiple of 2:
Favorable pairs: (2, 2), (2, 4), (2, 6) → 3 outcomes.

3. If the first roll x = 3, the second roll y must be a multiple of 3:
Favorable pairs: (3, 3), (3, 6) → 2 outcomes.

4. If the first roll x = 4, the second roll y must be a multiple of 4:
Favorable pairs: (4, 4) → 1 outcome.

5. If the first roll x = 5, the second roll y must be a multiple of 5:
Favorable pairs: (5, 5) → 1 outcome.

6. If the first roll x = 6, the second roll y must be a multiple of 6:
Favorable pairs: (6, 6) → 1 outcome.

Step 3: Count the total number of favorable outcomes
Summing up the favorable outcomes for each case:
$$\text{Total favorable outcomes}=6+3+2+1+1+1=14$$

Step 4: Calculate the probability
The probability P is the ratio of the number of favorable outcomes to the total number of possible outcomes:
$$P=\frac{14}{36}$$
Simplifying the fraction by dividing both the numerator and the denominator by their greatest common divisor, which is 2, we get:
$$P=\frac{7}{18}$$