Let's analyze the problem using the given data:

Given:

• Sample space points: 7 points.
• Probabilities of individual points:
  • $P(1) = P(2) = P(3) = P(7) = 0.1$,
  • $P(4) = P(5) = 0.05$,
  • $P(6) = 0.5$.
• Events $A$ and $B$ represented by a Venn diagram.

From the Venn diagram:

• $A = \{1, 2, 3, 5\}$,
• $B = \{2, 3, 4, 7\}$.

Now, solve each part:

(a) $P(A \cup B)$

The union of $A$ and $B$ is: $A \cup B = \{1, 2, 3, 4, 5, 7\}.$ The probability is: $P(A \cup B) = P(1) + P(2) + P(3) + P(4) + P(5) + P(7).$ Substituting: $P(A \cup B) = 0.1 + 0.1 + 0.1 + 0.05 + 0.05 + 0.1 = 0.5.$

(b) $P(B^c)$

The complement of $B$ is: $B^c = \{1, 5, 6\}.$ The probability is: $P(B^c) = P(1) + P(5) + P(6).$ Substituting: $P(B^c) = 0.1 + 0.05 + 0.5 = 0.65.$

(c) $P(B)$

From the Venn diagram: $B = \{2, 3, 4, 7\}.$ The probability is: $P(B) = P(2) + P(3) + P(4) + P(7).$ Substituting: $P(B) = 0.1 + 0.1 + 0.05 + 0.1 = 0.35.$

(d) $P(A \cup A^c)$

The union of $A$ and its complement ($A^c$) is the entire sample space. Hence: $P(A \cup A^c) = 1.$

Final Answers:

(a) $P(A \cup B) = 0.5$,
(b) $P(B^c) = 0.65$,
(c) $P(B) = 0.35$,
(d) $P(A \cup A^c) = 1$.

Would you like a deeper explanation or any step clarified?

Here are 5 related questions for further understanding:

1. What is the relationship between the union and intersection of two events?
2. How can we compute $P(A \cap B)$ from the given data?
3. How is the probability of $B^c$ always related to $P(B)$?
4. Can you explain why $P(A \cup A^c) = 1$ is always true?
5. How does the sample space affect the calculation of probabilities?

Tip: Always double-check whether the events in a probability problem overlap to avoid double counting.