Question

At a local high school with 200 female students, 98 play field hockey, 62 play volleyball, and 40 play both. Find the probability that a female student selected at random plays field hockey or volleyball. Write your answer as a simplified fraction. (1 point)
The probability that a female student
plays field hockey or volleyball is

Answer 1

To find the probability that a female student plays either field hockey or volleyball, we can use the principle of inclusion-exclusion to determine the number of students who play at least one of the two sports.

Let:
- \( A \) be the set of female students who play field hockey.
- \( B \) be the set of female students who play volleyball.

We have the following information:
- \( |A| = 98 \) (students who play field hockey)
- \( |B| = 62 \) (students who play volleyball)
- \( |A \cap B| = 40 \) (students who play both sports)

Using the principle of inclusion-exclusion, the number of students who play either field hockey or volleyball is given by:

\[
|A \cup B| = |A| + |B| - |A \cap B|
\]

Substituting the values:

\[
|A \cup B| = 98 + 62 - 40 = 120
\]

Now, we need to calculate the probability that a randomly chosen female student plays field hockey or volleyball:

\[
P(A \cup B) = \frac{|A \cup B|}{\text{Total number of female students}} = \frac{120}{200}
\]

To simplify this fraction:

\[
\frac{120}{200} = \frac{12}{20} = \frac{3}{5}
\]

Thus, the probability that a female student selected at random plays field hockey or volleyball is:

\[
\boxed{\frac{3}{5}}
\]