When a pair of fair dice is rolled, then the probability of getting the sum as multiple of 3 is

Answer

To find the probability of getting a sum that is a multiple of 3 when a pair of fair dice is rolled, we can follow these steps:

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

### Step 2: Identify the sums that are multiples of 3.
The possible sums when rolling two dice range from 2 (1+1) to 12 (6+6). The multiples of 3 within this range are:
- 3
- 6
- 9
- 12

### Step 3: Count the favorable outcomes for each multiple of 3.
Now, we will count the combinations that yield these sums:

- **Sum = 3**:
- (1, 2)
- (2, 1)

Total = 2 outcomes

- **Sum = 6**:
- (1, 5)
- (2, 4)
- (3, 3)
- (4, 2)
- (5, 1)

Total = 5 outcomes

- **Sum = 9**:
- (3, 6)
- (4, 5)
- (5, 4)
- (6, 3)

Total = 4 outcomes

- **Sum = 12**:
- (6, 6)

Total = 1 outcome

### Step 4: Calculate the total number of favorable outcomes.
Adding the favorable outcomes together:
\[ 2 \text{ (for sum 3)} + 5 \text{ (for sum 6)} + 4 \text{ (for sum 9)} + 1 \text{ (for sum 12)} = 12 \]

### Step 5: Calculate the probability.
The probability \( P \) of getting a sum that is a multiple of 3 is given by the ratio of the number of favorable outcomes to the total outcomes:
\[ P = \frac{\text{Number of favorable outcomes}}{\text{Total outcomes}} = \frac{12}{36} = \frac{1}{3} \]

### Final Answer:
The probability of getting the sum as a multiple of 3 when a pair of fair dice is rolled is \( \frac{1}{3} \).

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