Let's solve each part of the problem one by one.

### 1. **Listing the Sample Space**
The sample space \( S \) for tossing two dice consists of all possible pairs of outcomes. Since each die can land on any of the numbers 1 through 6, there are \( 6 \times 6 = 36 \) possible outcomes.

The sample space \( S \) can be listed as:
\[
S = \{(1,1), (1,2), (1,3), \ldots, (6,6)\}
\]

### 2. **Finding Probabilities**
Let's calculate the probabilities for the events \( A \), \( B \), and \( A \cap B \).

- **Event A**: The same number appears on both dice.
  
  The outcomes that satisfy this condition are:
  \[
  A = \{(1,1), (2,2), (3,3), (4,4), (5,5), (6,6)\}
  \]
  There are 6 favorable outcomes.

  So, the probability of \( A \) is:
  \[
  P(A) = \frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}} = \frac{6}{36} = \frac{1}{6}
  \]

- **Event B**: The sum of the two numbers is equal to 9.
  
  The outcomes that satisfy this condition are:
  \[
  B = \{(3,6), (4,5), (5,4), (6,3)\}
  \]
  There are 4 favorable outcomes.

  So, the probability of \( B \) is:
  \[
  P(B) = \frac{4}{36} = \frac{1}{9}
  \]

- **Event \( A \cap B \)**: The events \( A \) and \( B \) occur simultaneously. This means the same number appears on both dice and their sum is 9.
  
  The only possible outcome that satisfies both conditions is:
  \[
  A \cap B = \emptyset
  \]
  There are no favorable outcomes.

  So, the probability of \( A \cap B \) is:
  \[
  P(A \cap B) = \frac{0}{36} = 0
  \]

### 3. **Determining Independence**
Two events \( A \) and \( B \) are independent if:
\[
P(A \cap B) = P(A) \times P(B)
\]

Let's check:
\[
P(A) \times P(B) = \frac{1}{6} \times \frac{1}{9} = \frac{1}{54}
\]
Since \( P(A \cap B) = 0 \) and \( P(A) \times P(B) = \frac{1}{54} \), the events are **not independent**.

### 4. **Exploring Mutual Exclusivity**
Two events are mutually exclusive if they cannot occur at the same time, i.e., \( P(A \cap B) = 0 \).

Since \( P(A \cap B) = 0 \), events \( A \) and \( B \) are indeed **mutually exclusive**.

### Summary:
- \( P(A) = \frac{1}{6} \)
- \( P(B) = \frac{1}{9} \)
- \( P(A \cap B) = 0 \)
- Events \( A \) and \( B \) are not independent but are mutually exclusive.

Would you like more details on any part, or do you have any other questions?

### Follow-up Questions:
1. How would you calculate the probability of rolling a sum of 7 with two dice?
2. What is the probability of rolling two dice and getting different numbers?
3. If you roll two dice, what is the probability that the sum is an even number?
4. Can you explain the difference between mutually exclusive and independent events?
5. How would the probabilities change if the dice were not fair (e.g., one side is more likely to appear)?

### Tip:
When working with probabilities, always double-check whether the events are mutually exclusive or independent, as this greatly affects how probabilities are calculated.

Learn how to calculate probabilities for events A and B when tossing two dice. Includes step-by-step solutions for finding probabilities, checking for independence, and understanding mutual exclusivity. Suitable for Grades 9-12.

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