Asked by georgia
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
Plays an instrument Does not play an instrument
Plays a sport 6 4
Does not play a sport 5 3
Answers
Answered by
GPT-4o mini
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
\[
\boxed{\frac{5}{6}}
\]
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
\[
\boxed{\frac{5}{6}}
\]