Question

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?

	plays an instrument	does not play an instrument
does not play a sport	3	11

Explanation:

Step1: Calculate the total number of students who play an instrument

The number of students who play an instrument is the sum of those who play a sport and play an instrument and those who do not play a sport and play an instrument. So, \(8 + 3=11\).

Step2: Calculate the number of students who do not play a sport and play an instrument

From the table, this number is \(3\).

Step3: Use the formula for conditional probability

The formula for conditional probability \(P(A|B)=\frac{P(A\cap B)}{P(B)}\). In the context of "probability that a student does not play a sport given that they play an instrument", let \(A\) be "does not play a sport" and \(B\) be "plays an instrument". The formula simplifies to \(P = \frac{\text{Number of students who do not play a sport and play an instrument}}{\text{Number of students who play an instrument}}\). Substituting the values from Step1 and Step2, we get \(P=\frac{3}{11}\).

Answer:

\(\frac{3}{11}\)