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Let's break down the problem by analyzing the Venn diagram and solving each part:

### Definitions:
- \( A \) is the event "the student is in the Chess Club."
- \( B \) is the event "the student is in the Science Club."

The Venn diagram shows the memberships:
- **Chess only**: Raina, Charlie, Elsa, Lucy, Pablo, Carlos (6 students)
- **Both Chess and Science**: Josh, Heather, Jina (3 students)
- **Science only**: Lisa, Keith, Kelko, Ann (4 students)
- **Neither**: Leila (1 student)

### Total students: 14 (including Leila).

### (a) Find the probabilities of the events:
1. **\( P(A) \)**: Probability the student is in the Chess Club.
   - Students in the Chess Club: \(6 + 3 = 9\).
   - \( P(A) = \frac{9}{14} \).

2. **\( P(B) \)**: Probability the student is in the Science Club.
   - Students in the Science Club: \(4 + 3 = 7\).
   - \( P(B) = \frac{7}{14} \).

3. **\( P(A \text{ and } B) \)**: Probability the student is in both clubs.
   - Students in both clubs: Josh, Heather, Jina (3 students).
   - \( P(A \text{ and } B) = \frac{3}{14} \).

4. **\( P(A \mid B) \)**: Probability the student is in the Chess Club given they are in the Science Club.
   - This is calculated as \( P(A \text{ and } B) / P(B) = \frac{3}{14} \div \frac{7}{14} = \frac{3}{7} \).

5. **\( P(B) \cdot P(A \mid B) \)**: This is just \( P(A \text{ and } B) \).
   - \( P(B) \cdot P(A \mid B) = \frac{7}{14} \times \frac{3}{7} = \frac{3}{14} \).

### (b) Select the probability that is equal to \( P(A \text{ and } B) \):
- The correct answer is **\( P(B) \cdot P(A \mid B) \)** since it matches \( P(A \text{ and } B) \).

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Would you like further details or explanations?

### Related questions:
1. How would the probabilities change if a new student joined only the Science Club?
2. How would you calculate \( P(A \text{ or } B) \)?
3. What is the probability that a student is in neither club?
4. What is \( P(\text{not } A \mid B) \)?
5. Can the Venn diagram be used to find conditional probabilities in other contexts?

### Tip:
When working with Venn diagrams, always clearly define each section and count carefully to avoid over-counting or missing elements.

The Venn diagram below shows the 14 students in Ms. Stewart's class. The diagram shows the memberships for the Chess Club and the Science Club. A student from the class is randomly selected. Let A denote the event 'the student is in the Chess Club.' Let B denote the event 'the student is in the Science Club.' Answer questions regarding probabilities of these events and their intersections.

This problem involves calculating probabilities based on a Venn diagram of Chess Club and Science Club members in a class. It explores core probability concepts like intersections, unions, and conditional probability, and it is suitable for students in Grades 9-12.

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