Two dice are thrown at the same time. Determine the probability that the difference of the numbers on the two dice is :
(i) 0. (ii) 2.

Answer

To solve the problem of finding the probability that the difference of the numbers on two dice is 0 or 2, we will follow these steps:

### Step 1: Determine the total number of outcomes when two dice are thrown.
When two dice are thrown, each die has 6 faces. Therefore, the total number of outcomes is:
\[ \text{Total outcomes} = 6 \times 6 = 36 \]

### Step 2: Calculate the probability that the difference is 0.
The difference between the numbers on the two dice is 0 when both dice show the same number. The pairs that satisfy this condition are:
- (1, 1)
- (2, 2)
- (3, 3)
- (4, 4)
- (5, 5)
- (6, 6)

This gives us a total of 6 favorable outcomes. Thus, the probability \( P(\text{difference} = 0) \) is:
\[ P(\text{difference} = 0) = \frac{\text{Number of favorable outcomes}}{\text{Total outcomes}} = \frac{6}{36} = \frac{1}{6} \]

### Step 3: Calculate the probability that the difference is 2.
The difference between the numbers on the two dice is 2 if either die shows a number that is 2 greater than the other. The pairs that satisfy this condition are:
- (1, 3) and (3, 1)
- (2, 4) and (4, 2)
- (3, 5) and (5, 3)
- (4, 6) and (6, 4)

Counting these, we have:
- (1, 3)
- (3, 1)
- (2, 4)
- (4, 2)
- (3, 5)
- (5, 3)
- (4, 6)
- (6, 4)

This gives us a total of 8 favorable outcomes. Thus, the probability \( P(\text{difference} = 2) \) is:
\[ P(\text{difference} = 2) = \frac{\text{Number of favorable outcomes}}{\text{Total outcomes}} = \frac{8}{36} = \frac{2}{9} \]

### Summary of Results
- Probability that the difference is 0: \( \frac{1}{6} \)
- Probability that the difference is 2: \( \frac{2}{9} \)

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