Two dice are thrown simultaneously. Find the probability of getting an even number as the sum

Answer

To find the probability of getting an even number as the sum when two dice are thrown simultaneously, we can follow these steps:

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

### Step 2: Identify the favorable outcomes for an even sum
The possible sums when rolling two dice range from 2 (1+1) to 12 (6+6). The sums that are even are: 2, 4, 6, 8, 10, and 12. We need to find the combinations that yield these sums.

- **Sum = 2**: (1, 1)
- **Sum = 4**: (1, 3), (2, 2), (3, 1)
- **Sum = 6**: (1, 5), (2, 4), (3, 3), (4, 2), (5, 1)
- **Sum = 8**: (2, 6), (3, 5), (4, 4), (5, 3), (6, 2)
- **Sum = 10**: (4, 6), (5, 5), (6, 4)
- **Sum = 12**: (6, 6)

Now, let's count the favorable outcomes:
- For sum = 2: 1 outcome
- For sum = 4: 3 outcomes
- For sum = 6: 5 outcomes
- For sum = 8: 5 outcomes
- For sum = 10: 3 outcomes
- For sum = 12: 1 outcome

Adding these gives us:
\[ 1 + 3 + 5 + 5 + 3 + 1 = 18 \text{ favorable outcomes} \]

### Step 3: Calculate the probability
The probability \( P \) of getting an even sum is given by the ratio of the number of favorable outcomes to the total number of outcomes:
\[ P(\text{even sum}) = \frac{\text{Number of favorable outcomes}}{\text{Total outcomes}} = \frac{18}{36} = \frac{1}{2} \]

### Final Answer
The probability of getting an even number as the sum when two dice are thrown is:
\[ \frac{1}{2} \]

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