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 neither a sport nor 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 & 3 \\ \hline \begin{tabular}{c} Does not \\ play a \\ sport \end{tabular} & 11 & 5 \\ \hline \end{tabular}
To determine the probability that a student chosen randomly from the class plays neither a sport nor an instrument, we'll go through a step-by-step process:
1. Extract the given data from the table:
- Number of students who play both a sport and an instrument: 8 - Number of students who play a sport but do not play an instrument: 3 - Number of students who do not play a sport but play an instrument: 11 - Number of students who neither play a sport nor an instrument: 5
2. Calculate the total number of students in the class:
Sum all the students: [tex]\[
\text{Total students} = 8 + 3 + 11 + 5 = 27
\][/tex]
3. Identify the number of students who play neither a sport nor an instrument:
From the data, this number is given directly: [tex]\[
\text{Students who play neither sport nor instrument} = 5
\][/tex]
4. Calculate the probability:
The probability is given by the number of students who play neither a sport nor an instrument divided by the total number of students: [tex]\[
\text{Probability} = \frac{\text{Number of students who play neither sport nor instrument}}{\text{Total number of students}} = \frac{5}{27}
\][/tex]
5. Simplify the fraction (if possible):
Here, [tex]\(\frac{5}{27}\)[/tex] is already in its simplest form.
6. Convert the fraction to a decimal (if necessary):
So, the probability that a student chosen randomly from the class plays neither a sport nor an instrument is approximately [tex]\(0.1852 \text{ or } 18.52\% \)[/tex].
