The probability that the sum of the square of the two numbers, which show up when two fair dice are thrown, is even is

Options

A. `3/7`
B. `4/7`
C. `5/7`
D. None of these

Answer

To solve the problem of finding the probability that the sum of the squares of the two numbers shown when two fair dice are thrown is even, we can follow these steps:

### Step-by-Step Solution:

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

**Hint**: Remember that the total outcomes for rolling two dice is the product of the outcomes for each die.

2. **Identify Conditions for Even Sum**:
The sum of the squares of two numbers is even if:
- Both numbers are even (even + even = even)
- Both numbers are odd (odd + odd = even)

3. **Count Favorable Outcomes**:
- **Even Numbers on Dice**: The even numbers on a die are 2, 4, and 6.
- Possible pairs of even numbers: (2,2), (2,4), (2,6), (4,2), (4,4), (4,6), (6,2), (6,4), (6,6).
- Total pairs of even numbers = 9.

- **Odd Numbers on Dice**: The odd numbers on a die are 1, 3, and 5.
- Possible pairs of odd numbers: (1,1), (1,3), (1,5), (3,1), (3,3), (3,5), (5,1), (5,3), (5,5).
- Total pairs of odd numbers = 9.

- **Total Favorable Outcomes**:
\[ \text{Total Favorable Outcomes} = \text{Even Pairs} + \text{Odd Pairs} = 9 + 9 = 18 \]

**Hint**: When counting pairs, consider the combinations of even and odd numbers separately.

4. **Calculate Probability**:
The probability \( P \) that the sum of the squares is even is given by the ratio of favorable outcomes to total outcomes:
\[ P = \frac{\text{Favorable Outcomes}}{\text{Total Outcomes}} = \frac{18}{36} = \frac{1}{2} \]

**Hint**: Probability is always calculated as the ratio of favorable outcomes to total outcomes.

5. **Conclusion**:
The final answer is:
\[ \text{Probability} = \frac{1}{2} \]

### Final Answer:
The probability that the sum of the squares of the two numbers shown when two fair dice are thrown is even is \( \frac{1}{2} \).