To determine the probability that the first student chosen is a senior and the second student chosen is a sophomore, we need to follow these detailed steps:
1. Calculate the total number of students.
- Freshmen: 31
- Sophomores: 10
- Juniors: 17
- Seniors: 22
Therefore, the total number of students is:
[tex]\[
31 + 10 + 17 + 22 = 80
\][/tex]
2. Find the probability that the first student chosen is a senior.
To calculate this, we divide the number of seniors by the total number of students:
[tex]\[
\text{Probability (first student is a senior)} = \frac{22}{80} = 0.275
\][/tex]
3. Find the probability that the second student chosen is a sophomore, given that the students are chosen with replacement.
Since the student is chosen with replacement, the total number of students remains the same, and thus we calculate the probability of choosing a sophomore after the first draw independently:
[tex]\[
\text{Probability (second student is a sophomore)} = \frac{10}{80} = 0.125
\][/tex]
4. Calculate the combined probability of both events happening.
Since the events are independent (the first student is returned to the group), the combined probability is the product of the individual probabilities:
[tex]\[
\text{Combined probability} = 0.275 \times 0.125 = 0.034375
\][/tex]
5. Express the combined probability as a fraction.
The combined probability [tex]\(0.034375\)[/tex] can be converted into a fraction by recognizing that [tex]\(0.034375 = \frac{11}{320}\)[/tex].
Therefore, the probability that the first student chosen is a senior and the second student chosen is a sophomore is:
[tex]\[
\boxed{\frac{11}{320}}
\][/tex]
