Problem 52 - Precalculus with Limits

Question

The employees of a company work in six departments: \( 31 \) are in sales, \( 54 \) are in research, \( 42 \) are in marketing, \( 20 \) are in engineering,\( 47 \) are in finance, and \( 58 \) are in production. One employees paycheck is lost. What is the probability that the employee works in the research department?

Step-by-Step Solution

Answer

The probability that the employee works in the Research department is approximately 0.214 or 21.4%.

Step 1: Calculate Total Number of Employees

Add up the number of employees in each department to get the total number of employees. The total number is \(31 (Sales) + 54 (Research) + 42 (Marketing) + 20 (Engineering) + 47 (Finance) + 58 (Production) = 252\).

Step 2: Understand Basic Probability Concepts

The probability of an event is calculated by dividing the number of favorable outcomes by the total number of outcomes. This principle will be applied to this problem.

Step 3: Calculate the Probability

The favorable outcome is that the employee works in the Research department. There are 54 employees working in Research. So, the probability is calculated as \(P(Research Department) = \frac{Number of employees in Research}{Total number of employees} = \frac{54}{252} = 0.21428571428571427\).

Key Concepts

Basic Probability Concepts

Probability is a foundational concept in mathematics that helps us predict how likely an event is to occur. It is often expressed as a number between 0 and 1. A probability of 0 means the event cannot happen, while a probability of 1 means the event is certain to happen.
To calculate the probability of a single event, we use the formula:

• Probability = \( \frac{\text{Number of favorable outcomes}}{\text{Total number of possible outcomes}} \)

This formula allows us to measure the likelihood of events in various scenarios, such as determining the chance of picking a specific card from a deck or rolling a certain number on a die. Understanding basic probability concepts is crucial for assessing real-world situations.

Favorable Outcomes

In probability, a favorable outcome refers to the specific result or results we are interested in. They are called 'favorable' because they align with the outcome we wish to evaluate.
For example, in the given exercise, if we want to find out the probability of an employee working in the Research department, the favorable outcomes are the number of employees within that department. In this case, there are 54 employees in the Research department, making that the number of favorable outcomes.
Recognizing what counts as a favorable outcome is essential because this determines the numerator in our probability formula. It directly impacts the probability value, offering insights into how likely an event is.

Total Number of Employees

The total number of employees in this scenario forms the basis for the total number of possible outcomes. It is the key denominator in the probability formula. To find it, we sum up all employees from each department in the company.
Here's how it works for this problem:

• Sales: 31
• Research: 54
• Marketing: 42
• Engineering: 20
• Finance: 47
• Production: 58

Adding these values gives us the total of 252 employees. This total count of 252 is essential because it provides the complete set of possible outcomes that includes every employee in the company. Understanding this helps in making accurate probability calculations and understanding real-world organizational data.