Problem 3 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

List the simple events associated with each experiment. An opinion poll is conducted among a group of registered voters. Their political affiliationDemocrat (D), Republican ( \(R\) ), or Independent \((I)\) -and their sex-male \((m)\) or female \((f)-\) are recorded.

Step-by-Step Solution

Answer

The simple events associated with the opinion poll experiment are: 1. A registered male voter who is a Democrat (D, m) 2. A registered female voter who is a Democrat (D, f) 3. A registered male voter who is a Republican (R, m) 4. A registered female voter who is a Republican (R, f) 5. A registered male voter who is an Independent (I, m) 6. A registered female voter who is an Independent (I, f)

Step 1: Identify the possible outcomes for each variable.

First, we need to identify all the possible outcomes for each variable in this experiment: - Political affiliation: Democrat (D), Republican (R), and Independent (I) - Sex: male (m) and female (f)

Step 2: Create a list of simple events.

Now that we have the possible outcomes for each variable, we can list all the possible simple events by combining the outcomes for political affiliations and sex. These simple events can be represented as an ordered pair (abbreviation for political affiliation, abbreviation for sex): 1. (D, m) 2. (D, f) 3. (R, m) 4. (R, f) 5. (I, m) 6. (I, f)

Step 3: List the simple events associated with the experiment.

The simple events associated with the opinion poll experiment are: 1. A registered male voter who is a Democrat (D, m) 2. A registered female voter who is a Democrat (D, f) 3. A registered male voter who is a Republican (R, m) 4. A registered female voter who is a Republican (R, f) 5. A registered male voter who is an Independent (I, m) 6. A registered female voter who is an Independent (I, f) These simple events represent all the possible combinations of political affiliations and sex in the given opinion poll.

Key Concepts

Simple Events

In probability theory, a simple event is an outcome that cannot be further decomposed. It represents a single occurrence of some aspect of an experiment. For the opinion poll example, each combination of political affiliation and sex is considered a simple event.
This is because each of these combinations indicates a specific, indivisible result.

Consider that each voter can only belong to one political affiliation and one gender category. Thus, these combinations form the base units of our probability study in this example.

• A Democratic male, represented as (D, m)
• A Democratic female, represented as (D, f)
• A Republican male, represented as (R, m)
• A Republican female, represented as (R, f)
• An Independent male, represented as (I, m)
• An Independent female, represented as (I, f)

Each of these is a simple event because no further subdivision is possible without changing the event's nature.

Sample Space

The sample space in probability theory is the set of all possible simple events or outcomes that can occur in an experiment. For the opinion poll, the sample space includes each possible combination of political affiliation and sex.
This means considering every potential way the variables can interplay.

The sample space provides a comprehensive view of all possible outcomes, helping to calculate probabilities accurately. In our example, the sample space consists of six simple events:

• (D, m)
• (D, f)
• (R, m)
• (R, f)
• (I, m)
• (I, f)

Understanding the sample space is crucial because it forms the foundation for probability calculations. By knowing all possible events, you can ascertain the likelihood of any specific outcome.

Combinatorial Analysis

Combinatorial analysis is a method used to find the number of possible combinations or arrangements in various scenarios, especially when dealing with discrete structures. This is useful in probability when determining all potential outcomes in a given experiment, like the opinion poll in our exercise.
The process involves systematically listing combinations, ensuring none are missed.

In our example, combinatorial analysis helps in identifying the total number of combinations by multiplying the choices available for each variable. Determining combinations here means:

• Political affiliations: Democrat, Republican, Independent (3 options)
• Sex: male, female (2 options)

Using combinatorial analysis, the total number of simple events is calculated by multiplying these options: \3 \( \times \) 2 = 6 distinct combinations.

This approach ensures each unique pairing of characteristics is accounted for, supporting a complete probability model.