Question

Of the 400 fressmen at Rockdale High School, 92 students are in band, 60 students are in chorus, and 20 students are in both band and chorus. If a student is chosen at random, the P(in chorus | in band) is 21.7%.


T.
True


F.
False

Answer 1

To determine if the statement "the P(in chorus | in band) is 21.7%" is true or false, we need to calculate the conditional probability \( P(\text{Chorus | Band}) \).

The formula for conditional probability is:

\[
P(A | B) = \frac{P(A \text{ and } B)}{P(B)}
\]

In this case:

- Let \( A \) be the event that a student is in chorus.
- Let \( B \) be the event that a student is in band.

We know:
- The total number of students in band = \( 92 \)
- The total number of students in chorus = \( 60 \)
- The number of students in both band and chorus = \( 20 \)

We need to find \( P(\text{Chorus | Band}) \):

1. \( P(A \text{ and } B) = 20 \) (students in both band and chorus)
2. \( P(B) = 92 \) (students in band)

Now we can calculate:

\[
P(\text{Chorus | Band}) = \frac{P(A \text{ and } B)}{P(B)} = \frac{20}{92}
\]

Calculating \( \frac{20}{92} \):

\[
\frac{20}{92} = \frac{5}{23} \approx 0.2174
\]

To express this as a percentage, we multiply by 100:

\[
0.2174 \times 100 \approx 21.74\%
\]

The statement claims that \( P(\text{Chorus | Band}) \) is 21.7%. Since \( 21.74\% \) is approximately \( 21.7\% \), we can conclude that:

**The statement is True.**

**True (T)**.