At IDNLearn.com, find answers to your most pressing questions from experts and enthusiasts alike. Get timely and accurate answers to your questions from our dedicated community of experts who are here to help you.

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 does not play a sport given that they play an instrument?

\begin{tabular}{|c|c|c|}
\hline
& \begin{tabular}{c}
Plays an \\
instrument
\end{tabular}
& \begin{tabular}{c}
Does not play \\
an \\
instrument
\end{tabular} \\
\hline
\begin{tabular}{c}
Plays a \\
sport
\end{tabular}
& 8
& 6 \\
\hline
\begin{tabular}{c}
Does not \\
play a \\
sport
\end{tabular}
& 3
& 11 \\
\hline
\end{tabular}


Sagot :

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].