Questions: 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 plays a sport given that they play an instrument? Plays an instrument Does not play an instrument --------- Plays a sport 5 6 Does not play a sport 4 3

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 plays a sport given that they play an instrument? Plays an instrument Does not play an instrument --------- Plays a sport 5 6 Does not play a sport 4 3
Transcript text: 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 plays 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 Plays a sport & 5 & 6 \\ \hline \begin{tabular}{c} Does not play \\ a sport \end{tabular} & 4 & 3 \\ \hline \end{tabular}
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Solution

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Solution Steps

Step 1: Understand the Problem

We are given a contingency table that shows the number of students who play an instrument and/or a sport. We need to find the probability that a student plays a sport given that they play an instrument.

Step 2: Identify the Relevant Data

From the table, we have the following data:

  • Students who play both a sport and an instrument: 5
  • Students who play an instrument (regardless of playing a sport or not): 5 (plays a sport) + 4 (does not play a sport) = 9
Step 3: Calculate the Conditional Probability

The probability that a student plays a sport given that they play an instrument is calculated using the formula for conditional probability:

\[ P(\text{Plays a sport} \mid \text{Plays an instrument}) = \frac{P(\text{Plays a sport and plays an instrument})}{P(\text{Plays an instrument})} \]

Substituting the values from the table:

\[ P(\text{Plays a sport} \mid \text{Plays an instrument}) = \frac{5}{9} \]

Final Answer

The probability that a student plays a sport given that they play an instrument is \(\boxed{\frac{5}{9}}\).

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