In a class of 30 students, 11 play an instrument and 9 play a sport. There are 2 students who play an instrument and also play a sport. What is the probability that a student chosen randomly from the class plays a sport or an instrument?

We can solve this using the principle of inclusion-exclusion for sets.

Let:
- \( A \) be the set of students who play an instrument.
- \( B \) be the set of students who play a sport.

We are given the following:
- \( |A| = 11 \) (number of students who play an instrument),
- \( |B| = 9 \) (number of students who play a sport),
- \( |A \cap B| = 2 \) (number of students who play both an instrument and a sport).

We need to find the number of students who play either an instrument or a sport, which is the union of sets \( A \) and \( B \). The formula for the union of two sets is:

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

Substitute the given values:

\[
|A \cup B| = 11 + 9 - 2 = 18
\]

So, 18 students play either an instrument, a sport, or both.

Now, the total number of students in the class is 30. The probability that a randomly chosen student plays either an instrument or a sport is the ratio of students who play an instrument or sport to the total number of students:

\[
P(\text{plays a sport or instrument}) = \frac{|A \cup B|}{\text{Total students}} = \frac{18}{30} = \frac{3}{5}
\]

Thus, the probability that a randomly chosen student plays either a sport or an instrument is \( \frac{3}{5} \), or 0.6.

Would you like further details or have any questions?

Here are some related questions to explore:
1. What is the probability that a randomly chosen student plays neither a sport nor an instrument?
2. What is the probability that a student chosen at random plays both a sport and an instrument?
3. How would the probability change if 5 more students played only sports?
4. How many students play only an instrument?
5. How many students play only a sport?

**Tip:** To solve problems with multiple categories, inclusion-exclusion is a useful method to avoid double-counting.

Solve the probability problem where 11 students play an instrument, 9 play a sport, and 2 play both in a class of 30. Learn how to use the inclusion-exclusion principle to find that the probability a student plays a sport or instrument is 0.6. Suitable for Grades 6-8.

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