Question content area top

Question

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.

Answer 1

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: $$\frac{11}{14}$$