Question

A certain company plays classical music when its customers are on hold on the telephone. If the length of the complete recording, Mozart's Eine Kleine Nachtmusik is 2 hours long, what is the probability a customer put on hold will hear the Allegro movement which is 6 minutes, 31 seconds long?.

Answer 1

To determine the probability that a customer will hear the Allegro movement while on hold, we need to compare the duration of the Allegro movement to the total duration of the recording.

1. **Convert the lengths to the same unit (seconds)**:
- The total length of the recording is 2 hours:
\[
2 \text{ hours} = 2 \times 60 \times 60 = 7200 \text{ seconds}
\]
- The length of the Allegro movement is 6 minutes and 31 seconds:
\[
6 \text{ minutes} = 6 \times 60 = 360 \text{ seconds}
\]
\[
360 \text{ seconds} + 31 \text{ seconds} = 391 \text{ seconds}
\]

2. **Calculate the probability**:
The probability \( P \) that a customer will hear the Allegro movement is given by the ratio of the duration of the Allegro movement to the total duration of the recording.
\[
P = \frac{\text{Length of Allegro}}{\text{Total length of recording}} = \frac{391 \text{ seconds}}{7200 \text{ seconds}}
\]

3. **Perform the calculation**:
\[
P = \frac{391}{7200} \approx 0.05486
\]

4. **Convert to percentage (optional)**:
To express this probability as a percentage, multiply by 100:
\[
P \approx 0.05486 \times 100 \approx 5.486\%
\]

Thus, the probability that a customer put on hold will hear the Allegro movement is approximately \( \frac{391}{7200} \) or about \( 5.49\%\).