Problem 40 - Algebra for College Students

Question

A company has 500 employees among whom 200 are females, 15 are high-level executives, and 7 of the high-level executives are females. If one of the 500 employees is chosen at random, find the probability that the person chosen is female or is a high-level executive. \(0.416\)

Step-by-Step Solution

Answer

The probability is 0.416.

Step 1: Define the Sets

We define two sets: \( F \), which represents the females, and \( E \), which represents the high-level executives. So, \(|F| = 200\) and \(|E| = 15\).

Step 2: Identify the Intersection

Identify the employees who are both female and high-level executives. This is the intersection of \( F \) and \( E \), given as \(|F \cap E| = 7\).

Step 3: Use the Union Formula

Use the formula for the probability of the union of two events: \( P(F \cup E) = P(F) + P(E) - P(F \cap E) \).

Step 4: Calculate Each Probability

Calculate \( P(F) = \frac{|F|}{500} = \frac{200}{500} = 0.4 \), \( P(E) = \frac{|E|}{500} = \frac{15}{500} = 0.03 \), and \( P(F \cap E) = \frac{|F \cap E|}{500} = \frac{7}{500} = 0.014 \).

Step 5: Substitute and Solve

Substitute these values into the union formula: \( P(F \cup E) = 0.4 + 0.03 - 0.014 = 0.416 \).

Key Concepts

Union Formula

The Union Formula is a powerful tool in probability that helps us find the probability of either one event happening, or another, or both. In simpler terms, it tells us the likelihood of one event occurring or another.
The formula is expressed as:

• \( P(F \cup E) = P(F) + P(E) - P(F \cap E) \)

This means we add the probabilities of both events occurring individually, then subtract the probability of both events happening together, which accounts for any overlap. This overlap is known as the intersection of the events.
Here's how it can be applied:

• If we want to know the probability that an employee is either a female or a high-level executive, or both, we use this formula. We calculate the probability for each event separately and for their combination, then plug these numbers into the union formula.

Using the union formula makes solving complex probability problems much simpler, as it neatly accounts for any overlap between events.

Intersection of Sets

The Intersection of Sets is a fundamental concept in probability and set theory. It refers to the elements that are common to two or more sets.
If we think of sets as groups of items, the intersection is the group that belongs to more than one collection at the same time.
The intersection of two events \( F \) and \( E \) is noted as \( F \cap E \), which includes all elements present in both \( F \) and \( E \).

• In our exercise, the intersection \( |F \cap E| \) consists of the number of female employees who are also high-level executives. This is crucial because these employees are counted when calculating both standalone probabilities \( P(F) \) and \( P(E) \).

Recognizing the intersection helps prevent double-counting these employees when we want to find the probability of either event occurring.
This intersection value ensures precision in our probability calculations when we apply the Union Formula.

Probability of Events

Probability of events is the likelihood or chance that a particular outcome will happen. It’s one of the fundamental concepts needed to solve problems related to uncertainty and randomness.
In terms of sets, when we deal with probability,

• We examine subsets, like the set of all females \( F \), or high-level executives \( E \) within a larger set, here, the total employees.

For any event \( E \), the probability of \( E \) occurring is calculated as:

• \( P(E) = \frac{|E|}{|S|} \)

Where \( |E| \) represents the number of favorable outcomes, and \( |S| \) is the total number of possible outcomes.
In this context, \(|F|\) is the number of females, and \(|E|\) is the number of executives, while the entire sample set \(|S|\) is the total number of employees.

• This approach allows you to calculate the probability of specific groups, like the probability of randomly picking a female employee, a high-level executive, or a combination of both.

Understanding the probability of events is essential to solving problems that involve multiple groups or categories, giving a clearer picture of the odds involved.