Two dice are rolled simultaneously. Find the probability of :
getting a multiple of 3 as the sum.

Answer

To find the probability of getting a multiple of 3 as the sum when two dice are rolled, we can follow these steps:

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

### Step 2: Identify the possible sums
The possible sums when rolling two dice range from 2 (1+1) to 12 (6+6). The sums can be:
- 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12

### Step 3: Identify the multiples of 3 within the possible sums
The multiples of 3 within the range of possible sums (2 to 12) are:
- 3, 6, 9, 12

### Step 4: Count the favorable outcomes for each multiple of 3
Now we need to find the combinations that yield these sums:

- **Sum = 3**: (1,2), (2,1) → 2 outcomes
- **Sum = 6**: (1,5), (2,4), (3,3), (4,2), (5,1) → 5 outcomes
- **Sum = 9**: (3,6), (4,5), (5,4), (6,3) → 4 outcomes
- **Sum = 12**: (6,6) → 1 outcome

Now, let's add these favorable outcomes:
\[ 2 + 5 + 4 + 1 = 12 \text{ favorable outcomes} \]

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

### Final Answer
The probability of getting a multiple of 3 as the sum when two dice are rolled is:
\[ \frac{1}{3} \]

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