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)

Question

The probability that a female student plays field hockey or volleyball is
.

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 students who play field hockey.
• $B$ be the set of 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 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 the values:

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

Now, we find the probability that a randomly selected female student plays either field hockey or volleyball:

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

Now we simplify the fraction:

$$\frac{120}{200}=\frac{12}{20}=\frac{3}{5}$$

Thus, the probability that a female student plays field hockey or volleyball is

$$\boxed{{\displaystyle \frac{3}{5}}}.$$