To find the probability that the first student chosen is a senior and the second student chosen is a sophomore, we need to follow a detailed step-by-step approach:
1. Determine the Total Number of Students:
First, we sum up the number of students in all classes:
- Freshmen: 31
- Sophomores: 10
- Juniors: 17
- Seniors: 22
So, the total number of students is:
[tex]\[
31 + 10 + 17 + 22 = 80
\][/tex]
2. Probability of Selecting a Senior First:
The probability of selecting a senior first is the ratio of the number of seniors to the total number of students:
[tex]\[
\text{Probability of selecting a senior} = \frac{22}{80} = 0.275
\][/tex]
3. Probability of Selecting a Sophomore Second:
Since the student is replaced back into the group, the number of students remains the same for the second draw. Thus, the probability of selecting a sophomore second is:
[tex]\[
\text{Probability of selecting a sophomore} = \frac{10}{80} = 0.125
\][/tex]
4. Calculating the Combined Probability:
Since the events are independent (the student is replaced after the first draw), the combined probability is the product of the two individual probabilities:
[tex]\[
\text{Combined probability} = 0.275 \times 0.125 = 0.034375
\][/tex]
5. Converting the Probability to a Fraction:
Next, we convert the combined probability into a fraction and simplify it. The decimal 0.034375 can be expressed as a fraction:
[tex]\[
0.034375 = \frac{11}{320}
\][/tex]
6. Conclusion:
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]
Among the given answer choices, [tex]\(\frac{11}{320}\)[/tex], [tex]\(\frac{3}{80}\)[/tex], [tex]\(\frac{11}{40}\)[/tex], and [tex]\(\frac{2}{5}\)[/tex], the correct one is:
[tex]\[
\boxed{\frac{11}{320}}
\][/tex]
