Question

Question content area top
Part 1
In a certain Algebra and Trigonometry​ class, there are 20 male​ freshmen, 6 female​ freshmen, 12 male​ sophomores, and 13 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 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:

\[
\boxed{\frac{13}{17}}
\]