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Part 1
In a certain Algebra and Trigonometry class, there are 6 male freshmen, 13 female freshmen, 6 male sophomores, and 3 female sophomores. If a person is selected randomly from the group, find the probability that the selected person is a freshman or female.
Part 1
In a certain Algebra and Trigonometry class, there are 6 male freshmen, 13 female freshmen, 6 male sophomores, and 3 female sophomores. If a person is selected randomly from the group, find the probability that the selected person is a freshman or female.
Answers
GPT-4o mini
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:
\[
\frac{11}{14}
\]
### 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:
\[
\frac{11}{14}
\]