Question

a study of college business majors included 150 sophomores and 200 juniors. the study showed that 80 sophomores and 150 juniors had summer internships. one person from the study is selected at random. what is the probability that the person is a sophomore given that the person had a summer internship?
a. $\frac{8}{23}$
b. $\frac{3}{7}$
c. $\frac{8}{15}$
d. $\frac{23}{35}$

Explanation:

Step1: Calculate total interns

The number of sophomores with internships is 80 and the number of juniors with internships is 150. So the total number of students with internships is \(80 + 150=230\).

Step2: Apply conditional - probability formula

The formula for conditional probability \(P(A|B)=\frac{P(A\cap B)}{P(B)}\). In the case of counting problems, if \(A\) is the event of being a sophomore and \(B\) is the event of having an internship, then \(P(A|B)=\frac{n(A\cap B)}{n(B)}\), where \(n(A\cap B)\) is the number of sophomores with internships and \(n(B)\) is the total number of students with internships. Here, \(n(A\cap B) = 80\) and \(n(B)=230\). So the probability \(P=\frac{80}{230}=\frac{8}{23}\).

Answer:

A. \(\frac{8}{23}\)