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The image you uploaded shows a probability tree diagram with a problem asking for the value of \( p \).

### Analysis:
The tree diagram illustrates a sequence of two events, with each branch representing the probability of the event. The tree shows the following probabilities:
- For the first branch:
  - \( \frac{1}{3} \) (top path)
  - \( \frac{2}{3} \) (bottom path)
  
- For the second branch:
  - \( \frac{1}{3} \) (both top paths of each event)
  - \( \frac{2}{3} \) (both bottom paths of each event)
  
The problem asks to determine the value of \( p \).

### Step-by-Step Solution:

1. **Identify possible paths and their probabilities**:
   - **Top-Top Path**: \( \frac{1}{3} \times \frac{1}{3} = \frac{1}{9} \)
   - **Top-Bottom Path**: \( \frac{1}{3} \times \frac{2}{3} = \frac{2}{9} \)
   - **Bottom-Top Path**: \( \frac{2}{3} \times \frac{1}{3} = \frac{2}{9} \)
   - **Bottom-Bottom Path**: \( \frac{2}{3} \times \frac{2}{3} = \frac{4}{9} \)

2. **Sum the probabilities**:
   - The total probability should add up to 1: \( \frac{1}{9} + \frac{2}{9} + \frac{2}{9} + \frac{4}{9} = 1 \).

3. **Determine \( p \)**:
   - \( p \) likely refers to one of these probabilities, or it might represent the probability of a specific event or outcome that sums up the results of these branches.

   Given the options (A: \( \frac{1}{8} \), B: 9, C: \( \frac{1}{9} \), D: \( \frac{2}{3} \)), the most likely match for the value of \( p \) based on the paths is:
   - **\( p = \frac{2}{9} \)** (option C), which corresponds to two of the paths (Top-Bottom or Bottom-Top).

Thus, the correct answer is **option C: \( \frac{2}{9} \)**.

Would you like further details or explanations? Here are some related questions to expand your understanding:

1. How do probability trees help in determining compound probabilities?
2. What is the total probability rule, and how does it apply here?
3. How can you use probability trees to compute the probability of at least one event occurring?
4. Can you explain the difference between dependent and independent events in the context of this tree?
5. How would the tree diagram change if the events were dependent?

**Tip:** When working with probability trees, always ensure that the sum of all possible outcomes equals 1, which validates the probabilities are correct.

Write down the value for p based on the probability tree diagram.

Explore how to calculate the value of p using a probability tree diagram. This problem involves key concepts in probability, including the multiplication rule for independent events and the total probability theorem. Ideal for students in Grades 10-12.

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