Question

In a class of 22 students, 13 play an instrument and 7 play a sport. There are 5 students who play an instrument and also play a sport. What is the probability that a student does not play a sport given that they play an instrument? Write your answer as a simplified fraction or as a decimal rounded to 3 decimal places.(1 point) The probability that a student does not play a sport given that they play an instrument is:

Answer 1

To find the probability that a student does not play a sport given that they play an instrument, we can use the following information provided in the problem:

- Total number of students: 22
- Students who play an instrument: 13
- Students who play a sport: 7
- Students who play both an instrument and a sport: 5

Let's denote:

- \( A \): The event that a student plays an instrument.
- \( B \): The event that a student plays a sport.

We're interested in the probability \( P(B^c | A) \), where \( B^c \) is the complement of event \( B \), meaning the student does not play a sport.

Using the formula for conditional probability:

\[
P(B^c | A) = \frac{P(B^c \cap A)}{P(A)}
\]

1. **Calculate \( P(A) \)**: The probability that a student plays an instrument:
\[
P(A) = \frac{13}{22}
\]

2. **Calculate \( P(B^c \cap A) \)**: This is the number of students who play an instrument but do not play a sport.
- From the 13 students who play an instrument, 5 also play a sport.
- Therefore, the number of students who play an instrument but do not play a sport is:
\[
13 - 5 = 8
\]
- Thus, the probability \( P(B^c \cap A) \) is:
\[
P(B^c \cap A) = \frac{8}{22}
\]

3. **Calculate \( P(B^c | A) \)**:
Substitute the values into the conditional probability formula:
\[
P(B^c | A) = \frac{P(B^c \cap A)}{P(A)} = \frac{8/22}{13/22} = \frac{8}{13}
\]

Therefore, the probability that a student does not play a sport given that they play an instrument is:

\[
\boxed{\frac{8}{13}}
\]

If you would prefer this as a decimal, it can be calculated as:

\[
\frac{8}{13} \approx 0.615
\]

Thus, rounded to three decimal places, it's:

\[
\boxed{0.615}
\]