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:

(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 $7$: $P(A^c \cap B) = P(4) + P(7) = 0.1 + 0.05 = 0.15$

(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: $P(A) = P(1) + P(2) + P(3) + P(5) = 0.05 + 0.05 + 0.05 + 0.1 = 0.25$

(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: $P(A^c) = P(4) + P(6) + P(7) = 0.1 + 0.6 + 0.05 = 0.75$

Summary of Results:

• (a) $P(A^c \cap B) = 0.15$
• (b) $P(A) = 0.25$
• (c) $P(B^c) = 0.8$
• (d) $P(A^c) = 0.75$

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!