Option A is right

UpStudy Free Solution:

To determine the probability that an applicant meets the GPA requirements or volunteers, we need to use the principle of inclusion-exclusion. The formula for the probability of either event A or event B occurring is:

\(P( A \cup B) = P( A) + P( B) - P( A \cap B) \)

Where:

- \(P( A) \) is the probability of meeting the GPA requirements.

- \(P( B) \) is the probability of volunteering for community service.

- \(P( A \cap B) \) is the probability of both meeting the GPA requirements and volunteering.

Let's calculate each of these probabilities:

1. Probability of meeting the GPA requirements (\(P( A) \)):

\(P( A) = \frac { \text { Number of students who meet GPA requirements} } { \text { Total number of applicants} } = \frac { 950} { 2000} = 0.475\)

2. Probability of volunteering for community service (\(P( B) \)):

\(P( B) = \frac { \text { Number of students who volunteer} } { \text { Total number of applicants} } = \frac { 600} { 2000} = 0.3\)

3. Probability of both meeting the GPA requirements and volunteering (\(P( A \cap B) \)):

\(P( A \cap B) = \frac { \text { Number of students who meet GPA requirements and volunteer} } { \text { Total number of applicants} } = \frac { 250} { 2000} = 0.125\)

Now, use the inclusion-exclusion principle to find the probability that an applicant meets the GPA requirements or volunteers:

\(P( A \cup B) = P( A) + P( B) - P( A \cap B) = 0.475 + 0.3 - 0.125 = 0.65\)

The probability that an applicant meets the GPA requirements or volunteers is 65%.

Given that some applicants volunteer and meet the GPA requirements, the events are not mutually exclusive. Therefore, the correct statement is:

A. Because some applicants volunteer and meet the GPA requirements, the events are not mutually exclusive. Thus, the probability is 65%.

Key Concepts:

1. Mutually Exclusive Events: Two events are mutually exclusive if they cannot occur at the same time. For instance, drawing a red card and a black card simultaneously from a single draw in a deck of cards are mutually exclusive events.

2. Inclusion-Exclusion Principle: This principle is used to calculate the probability of the union of two events. For events A and B, the probability that either A or B occurs is given by:

\(P( A \cup B) = P( A) + P( B) - P( A \cap B) \)

Understanding the relationship between mutually exclusive events and the inclusion-exclusion principle is crucial for solving probability problems accurately. For personalized assistance and detailed, step-by-step solutions to your probability questions, try UpStudy Probability Calculator. With comprehensive explanations and tailored support, UpStudy ensures you grasp every concept and excel in your studies. Discover the benefits of UpStudy today and enhance your learning experience!