Question

a private university is accepting applications for enrollment. out of 2,000 applicants, 950 meet the gpa requirements, 600 volunteer for community service, and 250 both meet the gpa requirements and volunteer. which statement correctly describes the probability that an applicant meets the gpa requirements or volunteers? because some applicants volunteer and meet the gpa requirements, the events are not mutually exclusive. thus, the probability is 77.5%. because some applicants volunteer and meet the gpa requirements, the events are mutually exclusive. thus, the probability is 65%. because no applicants volunteer and meet the gpa requirements, the events are mutually exclusive. thus, the probability is 77.5%. because some applicants volunteer and meet the gpa requirements, the events are not mutually exclusive. thus, the probability is 65%.

Explanation:

Step1: Recall probability formula for non - mutually exclusive events

The formula for $P(A\cup B)$ is $P(A\cup B)=P(A)+P(B)-P(A\cap B)$. Let $A$ be the event of meeting GPA requirements and $B$ be the event of volunteering. We have $n(A) = 950$, $n(B)=600$, $n(A\cap B)=250$ and $n(S)=2000$.

Step2: Calculate individual probabilities

$P(A)=\frac{n(A)}{n(S)}=\frac{950}{2000}$, $P(B)=\frac{n(B)}{n(S)}=\frac{600}{2000}$, $P(A\cap B)=\frac{n(A\cap B)}{n(S)}=\frac{250}{2000}$.

Step3: Calculate $P(A\cup B)$

$P(A\cup B)=\frac{950}{2000}+\frac{600}{2000}-\frac{250}{2000}=\frac{950 + 600-250}{2000}=\frac{1300}{2000}=0.65 = 65\%$. Since some applicants are in both categories (volunteer and meet GPA requirements), the events are not mutually exclusive.

Answer:

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