The Venn diagram below shows the 10 students in Ms. Bennett's class.
The diagram shows the memberships for the Dance Club and the Volleyball Club.
Note that "Trey" is outside the circles since he is not a member of either club.

One student from the class is randomly selected.
Let A denote the event "the student is in the Dance Club."
Let B denote the event "the student is in the Volleyball Club."

Answer each part below. If necessary, consult a list of formulas.

(a)	Find the probabilities of the events below.
Write each answer as a single fraction.
P(A) =
P(B) =
P(A or B) = 
P(A and B) = 
P(A) + P(B) - P(A and B) =

 (b)	Select the probability that is equal to P(A)+P(B)-P(A and B)
P(B) 
P(A or B)
P(A and B)
P(A)


To answer this, we need to use basic probability rules and Venn diagram principles.

**Step-by-step solution:**

### **(a) Find the probabilities of the events:**

- **Total students in the class:**  
  There are 10 students in Ms. Bennett's class.

- Let’s define the quantities needed for each event from the Venn diagram:

1. **\( P(A) \): Probability of a student being in the Dance Club.**
    - Number of students in the Dance Club (including those in both clubs) = \( |A| \).
   
2. **\( P(B) \): Probability of a student being in the Volleyball Club.**
    - Number of students in the Volleyball Club (including those in both clubs) = \( |B| \).

3. **\( P(A \text{ or } B) \): Probability of a student being in either the Dance Club, the Volleyball Club, or both (union of A and B).**
    - The number of students who are in either Dance, Volleyball, or both clubs.

4. **\( P(A \text{ and } B) \): Probability of a student being in both the Dance Club and the Volleyball Club (intersection of A and B).**
    - Number of students in both the Dance and Volleyball Clubs.

5. **\( P(A) + P(B) - P(A \text{ and } B) \):**
    - This is simply the sum of the probabilities of being in the Dance Club and Volleyball Club minus the probability of being in both clubs (since we don't want to double-count those in both).

Now, we’ll use these steps to fill in the specific probabilities based on the data from the Venn diagram.

1. **\( P(A) \)**: Number of students in the Dance Club divided by total students.
2. **\( P(B) \)**: Number of students in the Volleyball Club divided by total students.
3. **\( P(A \text{ or } B) \)**: Number of students in Dance or Volleyball (including both) divided by total students.
4. **\( P(A \text{ and } B) \)**: Number of students in both Dance and Volleyball divided by total students.

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### **(b) Select the probability that is equal to \( P(A) + P(B) - P(A \text{ and } B) \)**:

This quantity simplifies to the formula for the union of two events. Therefore, the probability that equals this expression is \( P(A \text{ or } B) \).

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I can complete the specific probabilities once you provide the Venn diagram data (like how many students are in each section of the diagram). Do you want more details or have any questions?

### Related questions:
1. How do we calculate \( P(A \cap B) \) using the Venn diagram?
2. What is the difference between \( P(A \text{ or } B) \) and \( P(A \cap B) \)?
3. If Trey is outside both circles, how does this affect the total probability?
4. How does the addition rule work for non-mutually exclusive events?
5. How do we interpret the complement of \( P(A) \) in this context?

### Tip:
When calculating probabilities using a Venn diagram, always ensure that overlapping regions (intersections) are accounted for correctly to avoid overcounting.

The Venn diagram below shows the 10 students in Ms. Bennett's class. The diagram shows the memberships for the Dance Club and the Volleyball Club. Note that 'Trey' is outside the circles since he is not a member of either club. One student from the class is randomly selected. Let A denote the event 'the student is in the Dance Club.' Let B denote the event 'the student is in the Volleyball Club.' Answer each part below. If necessary, consult a list of formulas. (a) Find the probabilities of the events below. Write each answer as a single fraction. P(A) = P(B) = P(A or B) = P(A and B) = P(A) + P(B) - P(A and B) = (b) Select the probability that is equal to P(A)+P(B)-P(A and B) P(B) P(A or B) P(A and B) P(A)

Understand how to calculate probabilities using a Venn diagram. Solve the probability of events involving memberships in Dance and Volleyball clubs for Ms. Bennett's class. Learn about the addition rule of probability and how to handle intersections in probability problems. Suitable for Grades 7-9.

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