Bot GPT 4-o mini To find the probability that the selected person is a freshman or female, we can use the principle of inclusion-exclusion in probability.
First, let's determine the total number of students in the class:
- Male freshmen: 20
- Female freshmen: 6
- Male sophomores: 12
- Female sophomores: 13
Total number of students: \[ 20 + 6 + 12 + 13 = 51 \]
Next, we define the events:
- Let \( A \) be the event that the selected person is a freshman.
- Let \( B \) be the event that the selected person is female.
Calculating the number of freshmen (event \( A \)):
- Total freshmen = Male freshmen + Female freshmen = \( 20 + 6 = 26 \)
Calculating the number of females (event \( B \)):
- Total females = Female freshmen + Female sophomores = \( 6 + 13 = 19 \)
Next, we need to calculate the number of students who are both freshmen and female (the intersection of events \( A \) and \( B \)):
- Freshman females = 6 (as we have already counted above).
Now, we can use the formula for the union of two sets: \[ P(A \cup B) = P(A) + P(B) - P(A \cap B) \]
Calculating the probabilities:
- \( P(A) = \frac{\text{Number of freshmen}}{\text{Total number of students}} = \frac{26}{51} \)
- \( P(B) = \frac{\text{Number of females}}{\text{Total number of students}} = \frac{19}{51} \)
- \( P(A \cap B) = \frac{\text{Number of freshman females}}{\text{Total number of students}} = \frac{6}{51} \)
Putting these into the formula: \[ P(A \cup B) = \frac{26}{51} + \frac{19}{51} - \frac{6}{51} \]
Now, we compute \( P(A \cup B) \): \[ P(A \cup B) = \frac{26 + 19 - 6}{51} = \frac{39}{51} \]
Now, we can simplify \( \frac{39}{51} \): \[ \frac{39}{51} = \frac{13}{17} \]
Therefore, the probability that the selected person is a freshman or female is:
\[ \boxed{\frac{13}{17}} \]
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