To find the probability that a randomly selected person from the group is either a freshman or female, we need to count the total number of males and females in the class, as well as how many of those are freshmen.
Step 1: Determine total number of students
We can break the counts down as follows:
- Male Freshmen: 6
- Female Freshmen: 13
- Male Sophomores: 6
- Female Sophomores: 3
Total calculation:
- Total number of students = Male Freshmen + Female Freshmen + Male Sophomores + Female Sophomores
- Total number of students = 6 + 13 + 6 + 3 = 28
Step 2: Count the favorable outcomes
Next, we need to find the number of students who are either freshmen or female.
Counting freshmen:
- Total Freshmen = Male Freshmen + Female Freshmen = 6 + 13 = 19
Counting females:
- Total Females = Female Freshmen + Female Sophomores = 13 + 3 = 16
Step 3: Apply the principle of inclusion-exclusion
Some students are counted in both freshmen and females, specifically the female freshmen. We must be careful not to double-count them.
Female Freshmen:
- Female Freshmen = 13
Using the principle of inclusion-exclusion: \[ P(Freshman \cup Female) = P(Freshman) + P(Female) - P(Freshman \cap Female) \]
Calculating those probabilities:
- Number of freshmen = 19
- Number of females = 16
- Number of female freshmen = 13
Step 4: Calculate the total number of favorable outcomes:
Total number of favorable outcomes: \[ 19 + 16 - 13 = 22 \]
Step 5: Calculate the probability
Now, we can calculate the probability: \[ P(Freshman \cup Female) = \frac{\text{Number of favorable outcomes}}{\text{Total number of students}} \] \[ P(Freshman \cup Female) = \frac{22}{28} = \frac{11}{14} \]
Final Result
So the probability that a randomly selected person is either a freshman or female is: \[ \frac{11}{14} \]
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