1. **Problem 14:** Given $P(A)=0.4$, $P(B)=0.7$, and $P(A \cup B)=0.8$, find: a) $P(A \cup B')$ b) $P(A' \cap B)$ 2. **Step 1:** Recall that $B'$ is the complement of $B$, so $P(B')=1-P(B)=1-0.7=0.3$. 3. **Step 2:** Use the formula for union with complements: $$P(A \cup B') = P(A) + P(B') - P(A \cap B')$$ We need $P(A \cap B')$. 4. **Step 3:** Use the formula for $P(A \cup B)$: $$P(A \cup B) = P(A) + P(B) - P(A \cap B)$$ Substitute known values: $$0.8 = 0.4 + 0.7 - P(A \cap B) \implies P(A \cap B) = 0.4 + 0.7 - 0.8 = 0.3$$ 5. **Step 4:** Since $A = (A \cap B) \cup (A \cap B')$ and these are disjoint, $$P(A) = P(A \cap B) + P(A \cap B')$$ So, $$0.4 = 0.3 + P(A \cap B') \implies P(A \cap B') = 0.1$$ 6. **Step 5:** Now calculate $P(A \cup B')$: $$P(A \cup B') = P(A) + P(B') - P(A \cap B') = 0.4 + 0.3 - 0.1 = 0.6$$ 7. **Step 6:** For part b, $P(A' \cap B)$ is the part of $B$ not in $A$: $$P(A' \cap B) = P(B) - P(A \cap B) = 0.7 - 0.3 = 0.4$$ --- 8. **Problem 15:** 27 tourists asked about visits to Angola (A), Burundi (B), Cameroon (C). Given: - $|A|=15$, $|B|=8$, $|C|=12$ - $|A \cap B \cap C|=2$ - 21 visited only one country - Of those who visited Angola, 4 visited exactly one other country - Of those who did not visit Angola, 5 visited only Burundi - All visited at least one country 9. **Step a:** (Venn diagram) - Not drawable here but labels are as follows: - Only one country total: 21 tourists - $|A|=15$, with 4 visiting exactly one other country (so 4 in two-country intersections involving A) - $|B|=8$, with 5 visiting only Burundi and not Angola - $|C|=12$ - Triple intersection: 2 10. **Step b:** Find $|B'|$ and describe them. Since $B$ is the set of tourists who visited Burundi, $B'$ is those who did not visit Burundi. Total tourists = 27 $|B|=8$ so $$|B'| = 27 - 8 = 19$$ These 19 tourists did not visit Burundi. 11. **Step c:** Describe $(A \cup B) \cap C'$ and find its size. - $C'$ are tourists who did not visit Cameroon. - $(A \cup B) \cap C'$ are those who visited Angola or Burundi but not Cameroon. 12. **Step d:** Find probability a randomly selected tourist visited at least two countries. - Total tourists = 27 - Number who visited only one country = 21 - So number who visited at least two countries = $27 - 21 = 6$ - Probability = $\frac{6}{27} = \frac{2}{9} \approx 0.222$