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Let's solve the given problem by analyzing the Venn diagram and the probabilities associated with the sample points.

### Given probabilities:
- \( P(1) = P(2) = P(3) = P(7) = 0.05 \)
- \( P(4) = P(5) = 0.1 \)
- \( P(6) = 0.6 \)

### Venn Diagram Interpretation:
- Set \( A \) consists of the sample points \( \{1, 2, 3, 5\} \).
- Set \( B \) consists of the sample points \( \{3, 4, 7\} \).
- The intersection \( A \cap B \) consists of sample point \( \{3\} \).

Now, let's solve each part:

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#### (a) \( P(A^c \cap B) \)

- \( A^c \) is the complement of \( A \), meaning all the points not in \( A \), which are \( \{4, 6, 7\} \).
- \( A^c \cap B \) means points that are in both \( A^c \) and \( B \), which are \( \{4, 7\} \).

Thus, the probability \( P(A^c \cap B) \) is the sum of the probabilities of points \( 4 \) and \( 7 \):
\[
P(A^c \cap B) = P(4) + P(7) = 0.1 + 0.05 = 0.15
\]

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#### (b) \( P(A) \)

- \( P(A) \) is the probability of all points in set \( A \), which are \( \{1, 2, 3, 5\} \).

Thus, the probability \( P(A) \) is:
\[
P(A) = P(1) + P(2) + P(3) + P(5) = 0.05 + 0.05 + 0.05 + 0.1 = 0.25
\]

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#### (c) \( P(B^c) \)

- \( B^c \) is the complement of \( B \), meaning all the points not in \( B \), which are \( \{1, 2, 5, 6\} \).

Thus, the probability \( P(B^c) \) is:
\[
P(B^c) = P(1) + P(2) + P(5) + P(6) = 0.05 + 0.05 + 0.1 + 0.6 = 0.8
\]

---

#### (d) \( P(A^c) \)

- \( A^c \) is the complement of \( A \), meaning all the points not in \( A \), which are \( \{4, 6, 7\} \).

Thus, the probability \( P(A^c) \) is:
\[
P(A^c) = P(4) + P(6) + P(7) = 0.1 + 0.6 + 0.05 = 0.75
\]

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### Summary of Results:
- (a) \( P(A^c \cap B) = 0.15 \)
- (b) \( P(A) = 0.25 \)
- (c) \( P(B^c) = 0.8 \)
- (d) \( P(A^c) = 0.75 \)

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Would you like a detailed explanation on any part or have any further questions?

Here are some related questions:
1. How would you compute \( P(A \cup B) \)?
2. What is the probability \( P(A \cap B^c) \)?
3. What is the total probability of the entire sample space?
4. Can we verify if events \( A \) and \( B \) are independent?
5. How would you represent conditional probability \( P(A | B) \)?

Tip: When dealing with Venn diagrams, remember that intersections and unions are essential for understanding overlapping probabilities!

A sample space contains 7 sample points and events A and B as seen in the Venn diagram. Given the probabilities: P(1) = P(2) = P(3) = P(7) = 0.05, P(4) = P(5) = 0.1, and P(6) = 0.6. Use the Venn diagram and the probabilities of the sample points to find: (a) P(A^c ∩ B), (b) P(A), (c) P(B^c), and (d) P(A^c).

Explore how to calculate probabilities using Venn diagrams with sample points in a sample space of 7 events. Understand key concepts such as set complements, intersections, and probability calculations for sets A and B. Suitable for Grades 10-12.

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