Question

In a class of students, the following data table summarizes how many students play an instrument or a sport. What is the probability that a student chosen randomly from the class plays a sport or an instrument?
Plays an instrument Does not play an instrument
Plays a sport 6 4
Does not play a sport 5 3

Answer 1

To find the probability that a student chosen randomly from the class plays a sport or an instrument, we first need to summarize the data and determine the total number of students.

Here is the breakdown based on the given table:

- Plays an instrument and plays a sport: 6 students
- Plays an instrument and does not play a sport: 5 students
- Does not play an instrument but plays a sport: 4 students
- Does not play an instrument and does not play a sport: 3 students

Let's calculate the total number of students:

\[
\text{Total students} = 6 + 4 + 5 + 3 = 18
\]

Next, we need to find out how many students play either a sport or an instrument.

Students who play a sport are:
- Plays a sport and plays an instrument: 6
- Does not play an instrument but plays a sport: 4

Total students who play a sport = \(6 + 4 = 10\)

Students who play an instrument are:
- Plays an instrument and plays a sport: 6
- Does not play a sport but plays an instrument: 5

Total students who play an instrument = \(6 + 5 = 11\)

Now we apply the principle of inclusion-exclusion to count the number of students who play either a sport or an instrument. We must add the number of students who play a sport and the number who play an instrument, then subtract those who play both (since they are counted twice).

\[
\text{Plays a sport or an instrument} = (\text{Plays a sport}) + (\text{Plays an instrument}) - (\text{Plays both})
\]

Substituting the values we found:

\[
\text{Plays a sport or an instrument} = 10 + 11 - 6 = 15
\]

Thus, there are 15 students who play a sport or an instrument.

To find the probability that a randomly chosen student plays a sport or an instrument:

\[
\text{Probability} = \frac{\text{Number of students who play a sport or an instrument}}{\text{Total number of students}} = \frac{15}{18}
\]

This fraction can be simplified by dividing both the numerator and denominator by 3:

\[
\frac{15}{18} = \frac{5}{6}
\]

Therefore, the probability that a student chosen randomly from the class plays a sport or an instrument is

\[
\boxed{\frac{5}{6}}
\]