Problem 19 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

In a survey of 120 consumers conducted in a shopping mall, 80 consumers indicated that they buy brand A of a certain product, 68 buy brand \(\mathrm{B}\), and 42 buy both brands. How many consumers participating in the survey buy a. At least one of these brands? b. Exactly one of these brands? c. Only brand \(\mathrm{A}\) ? d. None of these brands?

Step-by-Step Solution

Answer

a. 106 consumers buy at least one of these brands. b. 22 consumers buy exactly one of these brands. c. 38 consumers buy only brand A. d. 14 consumers buy none of these brands.

Step 1: a. At least one of these brands

To find the number of consumers who buy at least one of these brands, we will use the principle of inclusion-exclusion, which states that: \( |A \cup B| = |A| + |B| - |A \cap B| \) where |A| represents the number of consumers that buy brand A, |B| represents the number of consumers that buy brand B, and |A ∩ B| represents the number of consumers that buy both brands. We are given the following values: |A|=80 (consumers that buy brand A) |B|=68 (consumers that buy brand B) |A ∩ B|=42 (consumers that buy both brands) Now we can plug these values into the equation: \( |A \cup B| = 80 + 68 - 42 = 106 \) So, 106 consumers buy at least one of these brands.

Step 2: b. Exactly one of these brands

To find the number of consumers who buy exactly one of these brands, we can subtract the number of consumers who buy both brands from the number of consumers who buy at least one of these brands: Exactly one brand = |A ∪ B| - 2|A ∩ B| Plugging in the values we found before: Exact one brand = 106 - 2(42) = 106 - 84 = 22 So, 22 consumers buy exactly one of these brands.

Step 3: c. Only brand A

To find the number of consumers who buy only brand A, we can subtract the number of consumers who buy both brands from the total number of consumers who buy brand A: Only brand A = |A| - |A ∩ B| Plugging in the known values: Only brand A = 80 - 42 = 38 So, 38 consumers buy only brand A.

Step 4: d. None of these brands

To find the number of consumers who buy none of these brands, we can subtract the number of consumers who buy at least one of these brands from the total number of consumers surveyed (120): None of these brands = Total consumers - |A ∪ B| Plugging in the known values: None of these brands = 120 - 106 = 14 So, 14 consumers buy none of these brands.

Key Concepts

Inclusion-Exclusion Principle

The Inclusion-Exclusion Principle is a powerful tool used in set theory to accurately count the number of elements in the union of several sets. This principle helps avoid over-counting elements that are present in more than one set.
In simpler terms, if you have two sets, say set A and set B, some elements may belong to both sets. If you simply add the totals of each set, you’re counting the overlapping elements twice. The Inclusion-Exclusion Principle provides a formula to correct this by subtracting the overlap.
Mathematically, it's expressed as:

• \( |A \cup B| = |A| + |B| - |A \cap B| \)

Here, \(|A|\) is the number of elements in set A, \(|B|\) is the number of elements in set B, and \(|A \cap B|\) is the number of elements common to both sets. Using this concept, as shown in the survey example, allows us to determine how many people buy at least one of the two brands by adjusting for the overlap of consumers who buy both.

Consumer Survey Analysis

Consumer Survey Analysis is a process often used in marketing and business to gather data on consumer behavior and preferences. By conducting surveys, companies can better understand the interests and habits of consumers. In this particular case, surveying consumer purchasing patterns gives insight into the market share of different brands.
For example, when 120 consumers were surveyed about their shopping choices between two brands, the data helped answer several questions:

• How many consumers purchase at least one of the brands?
• How many consumers purchase only one specific brand?
• How many do not buy either brand?

Conducting such surveys involves collecting data and analyzing results through methods such as using the Inclusion-Exclusion Principle. This further allows for strategic decision-making to improve sales and marketing tactics based on consumer preferences.

Problem Solving with Sets

Sets provide a useful way to organize and solve problems, especially when dealing with groups of items or people. Problem-solving with sets involves identifying the different groups involved, understanding their relationships, and using tools like Venn diagrams or set equations to find solutions.
In our example, the questions posed are solved by considering consumer groups as sets. We use set theory to break down complex problems involving overlap and separation of data:

• At least one brand: Solved using the Inclusion-Exclusion Principle, identifying total overlap.
• Exactly one brand: Calculated by subtracting the overlapping section from the count of any engagement.
• Only one brand: Found by discerning exclusive non-overlapping members of a set.
• Neither brand: Determined by complementing the union within the entire surveyed group.

These strategies help you compartmentalize and address each aspect of consumer data, giving clarity and direction to solving practical everyday problems efficiently.