To find the probability that a randomly selected person is either a freshman or female, we will first need to determine the total number of students and how many of those students meet the criteria.
Step 1: Count the total number of students.
We have the following students:
- Male freshmen: 10
- Female freshmen: 5
- Male sophomores: 18
- Female sophomores: 11
Total number of students: \[ 10 \text{ (male freshmen)} + 5 \text{ (female freshmen)} + 18 \text{ (male sophomores)} + 11 \text{ (female sophomores)} = 44 \text{ students} \]
Step 2: Calculate the number of freshmen.
Freshmen are comprised of both male and female freshmen: \[ 10 \text{ (male freshmen)} + 5 \text{ (female freshmen)} = 15 \text{ freshmen} \]
Step 3: Calculate the number of females.
Females include both female freshmen and female sophomores: \[ 5 \text{ (female freshmen)} + 11 \text{ (female sophomores)} = 16 \text{ females} \]
Step 4: Use the principle of inclusion-exclusion.
To find the total number of students who are either freshmen or female, we must add the count of freshmen and females, and then subtract those who are both (since they are counted twice).
The count of both freshmen and females is just the female freshmen (since they fall into both categories): \[ 5 \text{ female freshmen} \]
Step 5: Apply inclusion-exclusion.
Total (freshmen or females): \[ 15 \text{ (freshmen)} + 16 \text{ (females)} - 5 \text{ (female freshmen)} = 26 \]
Step 6: Calculate the probability.
The probability \( P \) that the selected person is a freshman or female is given by the number of favorable outcomes (freshmen or female) divided by the total number of students: \[ P = \frac{\text{Number of favorable outcomes}}{\text{Total number of students}} = \frac{26}{44} \]
Step 7: Simplify the fraction.
\[ P = \frac{26 \div 2}{44 \div 2} = \frac{13}{22} \]
Thus, the probability that a randomly selected person is a freshman or female is: \[ \boxed{\frac{13}{22}} \]
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