To find the probability that a randomly selected person is a freshman or female, we can use the formula for the probability of the union of two events:
\[ P(A \text{ or } B) = P(A) + P(B) - P(A \text{ and } B) \]
Where:
- \( A \) is the event that the selected person is a freshman.
- \( B \) is the event that the selected person is female.
Step 1: Determine the total number of students.
- Male freshmen: 11
- Female freshmen: 15
- Male sophomores: 11
- Female sophomores: 15
Total number of students:
\[ 11 + 15 + 11 + 15 = 52 \]
Step 2: Calculate \( P(A) \) (the probability of selecting a freshman).
The total number of freshmen (male and female):
\[ 11 + 15 = 26 \]
So, the probability \( P(A) \) is:
\[ P(A) = \frac{26}{52} = \frac{1}{2} \]
Step 3: Calculate \( P(B) \) (the probability of selecting a female).
The total number of females (freshmen and sophomores):
\[ 15 + 15 = 30 \]
So, the probability \( P(B) \) is:
\[ P(B) = \frac{30}{52} = \frac{15}{26} \]
Step 4: Calculate \( P(A \text{ and } B) \) (the probability of selecting a freshman who is female).
The total number of female freshmen:
\[ 15 \]
So, the probability \( P(A \text{ and } B) \) is:
\[ P(A \text{ and } B) = \frac{15}{52} \]
Step 5: Use the formula to find \( P(A \text{ or } B) \).
Now, we substitute these values into the formula:
\[ P(A \text{ or } B) = P(A) + P(B) - P(A \text{ and } B) \]
\[ P(A \text{ or } B) = \frac{1}{2} + \frac{15}{26} - \frac{15}{52} \]
To do this calculation, we can convert \( \frac{1}{2} \) and \( \frac{15}{26} \) to have a common denominator of 52:
\[ \frac{1}{2} = \frac{26}{52} \quad \text{and} \quad \frac{15}{26} = \frac{30}{52} \]
Now we can substitute:
\[ P(A \text{ or } B) = \frac{26}{52} + \frac{30}{52} - \frac{15}{52} \]
Combining the fractions:
\[ P(A \text{ or } B) = \frac{26 + 30 - 15}{52} = \frac{41}{52} \]
Conclusion:
The probability that a randomly selected person is a freshman or female is
\[ \frac{41}{52} \] or approximately 0.7885, which is about 78.85%.
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