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Find the indicated probability. Round your answer to 6 decimal places when necessary.

When two balanced dice are rolled, there are 36 possible outcomes. Find the probability that either doubles are rolled or the sum of the dice is 8.

To solve the problem, we need to find the probability that either doubles are rolled or the sum of the dice is 8 when two balanced dice are rolled. There are 36 possible outcomes when two dice are rolled.

### Step 1: Count the number of doubles
Doubles occur when both dice show the same number. The possible doubles are:
- (1,1)
- (2,2)
- (3,3)
- (4,4)
- (5,5)
- (6,6)

So, there are 6 possible outcomes that are doubles.

### Step 2: Count the number of outcomes where the sum is 8
The combinations that result in a sum of 8 are:
- (2,6)
- (3,5)
- (4,4)
- (5,3)
- (6,2)

So, there are 5 possible outcomes that sum to 8.

### Step 3: Determine the intersection of doubles and sum 8
Notice that (4,4) is counted in both the doubles and the sum of 8. Therefore, it is part of the intersection of the two sets.

### Step 4: Use the inclusion-exclusion principle
To find the probability that either event occurs, we use the inclusion-exclusion principle. The formula is:

\[
P(A \cup B) = P(A) + P(B) - P(A \cap B)
\]

Where:
- \(P(A)\) is the probability of rolling doubles.
- \(P(B)\) is the probability of the sum being 8.
- \(P(A \cap B)\) is the probability of both rolling doubles and the sum being 8.

### Step 5: Calculate the probabilities
- \(P(A) = \frac{6}{36} = \frac{1}{6}\)
- \(P(B) = \frac{5}{36}\)
- \(P(A \cap B) = \frac{1}{36}\) (since only (4,4) is in both sets)

So:

\[
P(A \cup B) = \frac{1}{6} + \frac{5}{36} - \frac{1}{36} = \frac{6}{36} + \frac{5}{36} - \frac{1}{36} = \frac{10}{36} = \frac{5}{18}
\]

Thus, the probability that either doubles are rolled or the sum of the dice is 8 is \( \boxed{\frac{5}{18}} \).

Would you like more details or have any questions about this solution?

Here are some related questions to consider:
1. What is the inclusion-exclusion principle and how is it applied in probability?
2. How would the probability change if we were looking for a sum of 7 instead of 8?
3. How many outcomes result in a sum of 10 when rolling two dice?
4. What is the probability of rolling two dice and getting a sum less than 5?
5. Can you list all the possible outcomes when rolling two dice?
6. How would you calculate the probability of getting at least one double in three rolls of two dice?
7. What is the complement rule in probability and how can it be used in problems like this?
8. How would the probability change if the dice were biased?

**Tip:** Always consider whether events are mutually exclusive or not when applying probability rules, as this determines whether to add or subtract overlapping probabilities.

Find the probability of either rolling doubles or the sum of 8 when two balanced dice are rolled. Step-by-step solution using combinatorics and the inclusion-exclusion principle. Suitable for Grades 10-12.

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