Question

all freshmen, sophomores, juniors, and seniors attended a high school assembly. the total student attendance is shown in the table.
class | number of people
freshmen | 31
sophomores | 10
juniors | 17
seniors | 22
twice during the assembly, a student is chosen at random to assist with the presentation. after the first student has finished assisting, the student returns to the group and can be chosen a second time. what is the probability that the first student chosen is a senior and the second student chosen is a sophomore?
\\(\frac{11}{320}\\)
\\(\frac{3}{80}\\)
\\(\frac{11}{40}\\)
\\(\frac{2}{5}\\)

Explanation:

Step1: Calculate total number of students

First, we need to find the total number of students by adding the number of students in each class. The number of freshmen is 31, sophomores is 10, juniors is 17, and seniors is 22. So the total number of students \( N = 31 + 10 + 17 + 22 \).
\[
N=31 + 10+17 + 22=80
\]

Step2: Probability of choosing a senior first

The number of seniors is 22, so the probability of choosing a senior first, \( P(\text{senior})=\frac{\text{number of seniors}}{\text{total number of students}}=\frac{22}{80} \)

Step3: Probability of choosing a sophomore second

Since the student is returned to the group, the total number of students remains 80. The number of sophomores is 10, so the probability of choosing a sophomore second, \( P(\text{sophomore})=\frac{\text{number of sophomores}}{\text{total number of students}}=\frac{10}{80} \)

Step4: Probability of both events happening

Since the two events are independent (because the student is replaced), the probability that the first is a senior and the second is a sophomore is the product of the two probabilities.
\[
P = P(\text{senior})\times P(\text{sophomore})=\frac{22}{80}\times\frac{10}{80}=\frac{22\times10}{80\times80}=\frac{220}{6400}=\frac{11}{320}
\]

Answer:

\(\frac{11}{320}\) (corresponding to the first option)