To find the probability of rolling doubles or a sum of eight with two standard dice, we can break this problem down into two parts and then apply the principle of inclusion-exclusion.

1. Rolling Doubles:

   • The doubles are (1,1), (2,2), (3,3), (4,4), (5,5), and (6,6).
   • There are 6 outcomes.

2. Rolling a Sum of Eight:

   • The pairs that result in a sum of eight are (2,6), (3,5), (4,4), (5,3), and (6,2).
   • There are 5 outcomes.

3. Counting Overlap:

   • The only overlap between doubles and the sum of eight is (4,4).
   • There is 1 outcome in the overlap.

4. Applying Inclusion-Exclusion:

   • Total outcomes = Outcomes of doubles + Outcomes of sum of eight - Overlap
   • Total outcomes = 6 + 5 - 1 = 10

5. Total Possible Outcomes:

   • When rolling two dice, there are $6 \times 6 = 36$ possible outcomes.

6. Probability Calculation: $P(\text{doubles or sum of 8}) = \frac{\text{Total favorable outcomes}}{\text{Total possible outcomes}} = \frac{10}{36} = \frac{5}{18}$

Thus, the probability of rolling doubles or a sum of eight is $\frac{5}{18}$.

Would you like more details or have any questions? Here are some related questions you might find interesting:

1. What are the probabilities of other combinations with two dice?
2. How would the probabilities change with three dice?
3. What is the probability of rolling a sum greater than ten?
4. How do you calculate expected values with dice rolls?
5. What is the significance of independent events in probability?

Tip: Always consider the total possible outcomes when calculating probabilities to avoid errors.