Certainly! Let's break down the problem step-by-step to find the probability that a student does not play a sport given that they play an instrument.
### Step 1: Identify the Relevant Data
From the table:
- Students who play both a sport and an instrument: 8
- Students who play an instrument but do not play a sport: 3
- Students who do not play an instrument but play a sport: 6
- Students who neither play an instrument nor a sport: 11
### Step 2: Determine the Total Number of Students who Play an Instrument
The total number of students who play an instrument is the sum of students who play both a sport and an instrument and those who play only an instrument:
[tex]\[
\text{Total plays instrument} = 8 (\text{both}) + 3 (\text{only instrument}) = 11
\][/tex]
### Step 3: Find Out How Many Students do Not Play a Sport but Play an Instrument
From the table, the number of students who play an instrument but do not play a sport is given:
[tex]\[
\text{Does not play sport given instrument} = 3
\][/tex]
### Step 4: Calculate the Conditional Probability
The conditional probability that a student does not play a sport given that they play an instrument is the ratio of the number of students who play an instrument but do not play a sport to the total number of students who play an instrument.
[tex]\[
P(\text{Does not play sport} \mid \text{Plays instrument}) = \frac{\text{Number of students who do not play sport but play instrument}}{\text{Total number of students who play instrument}}
\][/tex]
So, substituting the numbers:
[tex]\[
P(\text{Does not play sport} \mid \text{Plays instrument}) = \frac{3}{11} \approx 0.273
\][/tex]
### Conclusion
The probability that a student does not play a sport given that they play an instrument is approximately [tex]\(0.273\)[/tex] or [tex]\(27.27\%\)[/tex].
