Question

A marketing class did a sample survey to find out how many of a class of 125 people owned CDs of the Beatles, Alabama, or Bob Marley. The results of the survey showed the following:$$\begin{array}{|l|c|}\hline \text { Recording Artist } & \text { No. of Students Owning CDs } \\ \hline \text { Beatles } & 65 \\\\\hline \text { Alabama } & 46 \\\\\hline \text { Bob Marley } & 29 \\\\\hline \text { Beatles and Alabama } & 18 \\\\\hline \text { Beatles and Bob Marley } & 21 \\\\\hline \text { Bob Marley and Alabama } & 12 \\\\\hline \text { Beatles, Bob Marley, and Alabama } & 9 \\\\\hline\end{array}$$How many of the students owned no CD featuring these performers?

Short answer

26 students do not own any CDs by the Beatles, Alabama, or Bob Marley.

Step-by-step solution

Understand the Problem

We need to determine the number of students who own no CDs by the Beatles, Alabama, or Bob Marley from a total of 125 students.

Use the Inclusion-Exclusion Principle

The Inclusion-Exclusion Principle helps us calculate the number of students who own at least one CD. The formula is: \[|A \cup B \cup C| = |A| + |B| + |C| - |A \cap B| - |A \cap C| - |B \cap C| + |A \cap B \cap C|\] where \(|A| = 65\), \(|B| = 46\), \(|C| = 29\), \(|A \cap B| = 18\), \(|A \cap C| = 21\), \(|B \cap C| = 12\), \(|A \cap B \cap C| = 9\).

Calculate Students Owning at least One CD

Substitute the given numbers into the formula to find the number of students who own at least one CD:\[\begin{align*}|A \cup B \cup C| & = 65 + 46 + 29 - 18 - 21 - 12 + 9 \& = 141 - 51 + 9 \& = 99.\end{align*}\] Thus, 99 students own at least one CD.

Calculate Students Owning No CDs

Subtract the number of students owning at least one CD from the total number of students:\[125 - 99 = 26.\] Therefore, 26 students do not own any CDs by these performers.

Key concepts

Set Theory

Set theory deals with the study of sets, which are collections of objects. In this context, each music artist (Beatles, Alabama, Bob Marley) is represented by a set that contains students owning their CDs. Key concepts in set theory include union, intersection, and complement.
- The **union** of sets (\(|A \cup B \cup C|\), for example) represents all distinct elements found in any of the sets.
- The **intersection** (\(|A \cap B|\) for example) includes only those elements present in all sets.
- When it comes to solutions, the **complement** shows elements not included in a set, beneficial when calculating those not owning any of the artists' CDs.
Set theory is fundamental for understanding how to efficiently categorize data and solve problems involving multiple categories or groups.

Survey Analysis

Survey analysis involves interpreting data gathered from a group of people to gain insights or answer specific questions. The exercise in question represents a classic survey analysis task where we determine how many students own CDs of certain musical artists. Key components in survey analysis include:
- **Data Collection**: Gathering results like the number of CDs owned by students.
- **Data Interpretation**: Understanding relationships and overlaps, such as students owning CDs from more than one artist.
- **Statistical Techniques**: Using tools like the Inclusion-Exclusion Principle to assess the data more accurately.
Survey analysis helps dissect collected data to extract meaningful insights and understand trends among the surveyed audience.

Problem Solving Steps

Solving problems efficiently often requires a systematic approach. Let's examine the steps taken to solve the given problem using the Inclusion-Exclusion Principle.
- **Step 1: Understand the Problem**: Identifying the need to find how many students own no artist CDs out of a total class of 125. Clarifying what is known and what needs solving is crucial.
- **Step 2: Apply the Inclusion-Exclusion Principle**: A powerful formula that helps find the total number of students owning at least one CD by considering and correcting for those counted multiple times across overlapping sets.
- **Step 3: Calculate Results**: By substituting the given survey numbers into the formula, you determine 99 students own at least one CD.
- **Step 4: Final Calculation**: Subtract 99 from the total 125 to find that 26 students own no CDs at all.
Breaking down problems into these clear, logical steps helps ensure a thorough understanding and accurate solution, reducing errors and increasing confidence in the solution reached.