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To solve the problem, let's break it down step by step using the given information:

---

### Problem:  
We need to calculate \( P(A \mid B) \), which is the probability of \( A \) given \( B \). This is defined as:  
\[
P(A \mid B) = \frac{P(A \cap B)}{P(B)}
\]

---

### Step 1: Identify the key values from the Venn diagram.

1. The number of items in \( A \cap B \) (overlap of \( A \) and \( B \)) is **3**.
2. The total number of items in \( B \) is the sum of the values in \( B \):
   \[
   P(B) = 3 + 5 = 8
   \]

---

### Step 2: Calculate \( P(A \cap B) \) and \( P(B) \).

The probability of \( A \cap B \) is the number of items in \( A \cap B \) divided by the total number of items in the universal set (\( \xi \)):
\[
P(A \cap B) = \frac{3}{24}
\]
The total number of items in the universal set is:
\[
9 + 7 + 3 + 5 = 24
\]

The probability of \( B \) is:
\[
P(B) = \frac{8}{24}
\]

---

### Step 3: Calculate \( P(A \mid B) \).

Using the formula \( P(A \mid B) = \frac{P(A \cap B)}{P(B)} \):
\[
P(A \mid B) = \frac{\frac{3}{24}}{\frac{8}{24}} = \frac{3}{8}
\]

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### Final Answer:
\[
P(A \mid B) = \frac{3}{8}
\]

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Would you like further details, or do you have any additional questions?  

### 5 Related Questions:
1. How is conditional probability \( P(A \mid B) \) different from \( P(A \cap B) \)?
2. Can we calculate \( P(B \mid A) \) using this diagram?
3. What would the probability of selecting an item outside both \( A \) and \( B \) be?
4. How can we verify the total probability of all outcomes in the universal set?
5. What is the complement rule, and how does it apply to this problem?

### Tip:
Always ensure the probabilities add up to 1 when summing across all mutually exclusive events in the universal set!

The Venn diagram below shows information about the number of items in sets A and B. An item is chosen at random. Calculate P(A|B). Give your answer as a fraction.

Learn how to calculate conditional probability P(A|B) using a Venn diagram. This problem involves determining probabilities from set data, calculating intersections, and applying the formula P(A|B) = P(A ∩ B) / P(B). Suitable for Grades 9-10.

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