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