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 \( 5.49%\).