To find the probability that the person is a sophomore given that the person had a summer internship, we'll work through the problem step-by-step:
1. Identify the Given Information:
- Total sophomores: 150
- Total juniors: 200
- Sophomores with internships: 80
- Juniors with internships: 150
2. Calculate the Total Number of Students with Internships:
We add the number of sophomores with internships and juniors with internships:
[tex]\[
\text{Total with internships} = 80 + 150 = 230
\][/tex]
3. Set Up the Probability Formula:
We need to find the probability that a randomly selected student is a sophomore given that they had a summer internship. This probability can be denoted as [tex]\( P(\text{Sophomore} \mid \text{Internship}) \)[/tex].
The formula for conditional probability is:
[tex]\[
P(\text{Sophomore} \mid \text{Internship}) = \frac{P(\text{Sophomore} \cap \text{Internship})}{P(\text{Internship})}
\][/tex]
4. Calculate the Probability of Students Having an Internship:
[tex]\[
P(\text{Internship}) = \frac{\text{Total with internships}}{\text{Total number of students}} = \frac{230}{150 + 200} = \frac{230}{350}
\][/tex]
5. Calculate the Joint Probability of Being a Sophomore and Having an Internship:
Since we already know the number of sophomores with internships, we can write:
[tex]\[
P(\text{Sophomore} \cap \text{Internship}) = \frac{\text{Sophomores with internships}}{\text{Total number of students}} = \frac{80}{350}
\][/tex]
6. Use the Joint Probability and the Total Probability of Having an Internship to Find the Conditional Probability:
[tex]\[
P(\text{Sophomore} \mid \text{Internship}) = \frac{P(\text{Sophomore} \cap \text{Internship})}{P(\text{Internship})} = \frac{\frac{80}{350}}{\frac{230}{350}} = \frac{80}{230}
\][/tex]
7. Simplify the Fraction:
To simplify the fraction [tex]\(\frac{80}{230}\)[/tex], we find the greatest common divisor (GCD) of 80 and 230. The GCD is 10.
[tex]\[
\frac{80}{230} = \frac{80 \div 10}{230 \div 10} = \frac{8}{23}
\][/tex]
Thus, the probability that the person is a sophomore given that the person had a summer internship is:
[tex]\[
\boxed{\frac{8}{23}}
\][/tex]
