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Let's break down the problem step by step:

### General Information
Each die has 6 faces, so when rolling two dice, the number of possible outcomes is:

\[
6 \times 6 = 36
\]

We can now answer the parts of the question based on this.

### Part (a): Probability of rolling a sum not more than 8
We need to find the number of outcomes where the sum of the dice is less than or equal to 8. The possible sums and their corresponding outcomes are:

- Sum = 2: (1,1)
- Sum = 3: (1,2), (2,1)
- Sum = 4: (1,3), (2,2), (3,1)
- Sum = 5: (1,4), (2,3), (3,2), (4,1)
- Sum = 6: (1,5), (2,4), (3,3), (4,2), (5,1)
- Sum = 7: (1,6), (2,5), (3,4), (4,3), (5,2), (6,1)
- Sum = 8: (2,6), (3,5), (4,4), (5,3), (6,2)

The total number of favorable outcomes is \(1 + 2 + 3 + 4 + 5 + 6 + 5 = 26\).

The probability is then:
\[
P(\text{sum} \leq 8) = \frac{26}{36} = \frac{13}{18}
\]

### Part (b): Probability of rolling a sum not less than 11
We need to count the number of outcomes where the sum of the dice is greater than or equal to 11:

- Sum = 11: (5,6), (6,5)
- Sum = 12: (6,6)

There are 3 favorable outcomes, so the probability is:
\[
P(\text{sum} \geq 11) = \frac{3}{36} = \frac{1}{12}
\]

### Part (c): Probability of rolling a sum between 6 and 11 (exclusive)
We need to find the number of outcomes where the sum is strictly between 6 and 11. That means the sum can be 7, 8, 9, or 10. The outcomes are:

- Sum = 7: (1,6), (2,5), (3,4), (4,3), (5,2), (6,1)
- Sum = 8: (2,6), (3,5), (4,4), (5,3), (6,2)
- Sum = 9: (3,6), (4,5), (5,4), (6,3)
- Sum = 10: (4,6), (5,5), (6,4)

The total number of favorable outcomes is \(6 + 5 + 4 + 3 = 18\).

Thus, the probability is:
\[
P(6 < \text{sum} < 11) = \frac{18}{36} = \frac{1}{2}
\]

### Final answers:
- (a) \(\frac{13}{18}\)
- (b) \(\frac{1}{12}\)
- (c) \(\frac{1}{2}\)

Let me know if you'd like further details or clarification! Here are some related questions for practice:

1. What is the probability of rolling a sum of exactly 7?
2. What is the probability of rolling a sum of exactly 5?
3. What is the probability of rolling a sum greater than 10?
4. How many ways can you roll a sum of 9?
5. What is the probability of rolling doubles (e.g., (1,1), (2,2), etc.)?

**Tip:** When calculating probabilities with dice, listing all possible outcomes can help visualize patterns!

A pair of dice is rolled. Find the probability of rolling: (a) a sum not more than 8, (b) a sum not less than 11, (c) a sum between 6 and 11 (exclusive).

Learn how to calculate the probability of rolling specific sums with two dice. This problem covers calculating the probability for sums not more than 8, sums not less than 11, and sums between 6 and 11 (exclusive). Step-by-step solutions and explanations included. Suitable for middle school students.

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