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The document contains a series of questions related to probability, set theory, and surveys. Let's go through some of the solutions step-by-step.

### Question 1
#### (a) Economic Survey of 300 Households
1. **Probability a household does not prefer Product B:**
   - Total households = 300
   - Households preferring Product B = 50
   - Households not preferring Product B = 300 - 50 = 250
   - Probability = 250 / 300 = 5/6.

2. **Probability a household prefers neither Product A nor Product B:**
   - Households preferring A or B or both = 70 + 50 - 20 = 100.
   - Households preferring neither = 300 - 100 = 200.
   - Probability = 200 / 300 = 2/3.

#### (b) Services Usage Study of 382 Businesses
1. **Businesses using Service X and Z but not Y:**
   - \( n(X \cap Z) = 40 \)
   - \( n(X \cap Z \cap \neg Y) = n(X \cap Z) - n(X \cap Z \cap Y) = 40 - 20 = 20 \).

2. **Businesses using Service Y and Z but not X:**
   - \( n(Y \cap Z) = 36 \)
   - \( n(Y \cap Z \cap \neg X) = n(Y \cap Z) - n(X \cap Y \cap Z) = 36 - 20 = 16 \).

3. **Businesses using Service Y but neither Service X nor Z:**
   - Total using Y = 152
   - \( n(Y \cap \neg X \cap \neg Z) = n(Y) - (n(X \cap Y) + n(Y \cap Z) - n(X \cap Y \cap Z)) = 152 - (72 + 36 - 20) = 64 \).

4. **Businesses using none of the three services:**
   - Total = 382
   - Using at least one = \( 382 - n(\neg X \cap \neg Y \cap \neg Z) = 382 - n(\neg X \cup \neg Y \cup \neg Z) \).
   - This requires applying the inclusion-exclusion principle.

#### (c) Preferences for Economic Policies
Using the formula for inclusion-exclusion and solving for the percentage who favor all three policies, based on given data for pairwise combinations.

### Question 2
#### (a) Symmetric Difference
- \( A = \{a, e, r, 1, f, 2, g, 3, h\} \)
- \( B = \{1, a, 6, b, 5, 9, c, d, w, r\} \)
- \( A \Delta B = (A \cup B) - (A \cap B) \).

#### (b) Company Concerns
Applying the principle of inclusion-exclusion to find the number of companies surveyed, those with at least one concern, and specific ones with certain concerns.

### Question 3
#### (b) Venn Diagram of Economists
1. Construct a Venn diagram to visualize the overlaps.
2. Calculate the number of economists who learned about the policy from at least two sources using overlaps.
3. Determine those who learned from exactly one source by subtracting those learning from two or more.

For any specific calculation or further explanations, let me know! Here are some related questions that can deepen your understanding:

1. How can the inclusion-exclusion principle be applied in probability problems involving multiple sets?
2. What are some practical applications of set theory in business and economics?
3. How can Venn diagrams assist in visualizing complex relationships among data sets?
4. What techniques can be used to find probabilities in overlapping events?
5. How do probability and set theory concepts integrate into decision-making processes?

**Tip:** Always verify set relations using Venn diagrams, as they can simplify complex relationships and help avoid errors in logic.

The document contains a series of probability and set theory problems. Here are some of the questions:
1. Probability problems based on preferences for Product A and Product B in a group of 300 households.
2. Set theory problems involving a study of 382 businesses using Services X, Y, and Z.
3. Probability calculations for economic policy preferences in a survey, using percentage overlaps.
4. Symmetric difference and set operations on defined sets.
5. Problems involving Venn diagrams for career preferences among students and economists.

This page provides detailed solutions to probability and set theory problems, including calculating probabilities in economic surveys, analyzing business service usage with Venn diagrams, and determining overlaps in career preferences. Suitable for Grades 11-12 or undergraduate students, the solutions cover key mathematical concepts such as the inclusion-exclusion principle, probability calculations, and set operations.

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