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 i

Answer 1

To find the probability that a female student selected at random plays either field hockey or volleyball, we can use the principle of inclusion-exclusion.

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

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

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

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

Substituting in the known values:

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

Now, to find the probability that a randomly selected female student plays either field hockey or volleyball, we divide the number of students who play either sport by the total number of female students:

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

This simplifies to:

\[
P(A \cup B) = \frac{120 \div 40}{200 \div 40} = \frac{3}{5}
\]

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

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