Understanding the Probability of Three Telephone Lines Being in Use

In today's busy office environments, efficient communication is crucial. Let's consider an office that has three telephone lines. We know that at any given time, the probability that at least one of these lines is in use is 0.8. Our goal is to determine the probability that all three lines are in use at the same time.

Setting Up the Problem

First, let's define our probabilities clearly. Let:

PA be the probability that at least one line is in use, which is given as 0.8.

PB be the probability that all three lines are in use.

Using the given information, we know that the probability that at least one line is in use can be expressed as:

PA 1 - Pnone are in use

Therefore:

Pnone are in use 1 - 0.8 0.2

Assumptions and Calculations

We assume that the probability of each line being in use is independent. Let p be the probability that a single line is in use. The probability that none of the three lines are in use can be expressed as:

Pnone are in use (1 - p)3

Equating this to the previously calculated probability:

(1 - p)3 0.2

Taking the cube root of both sides, we get:

1 - p 0.21/3

Calculating 0.21/3:

0.21/3 ≈ 0.5848

Thus:

1 - p ≈ 0.5848

This implies:

p ≈ 1 - 0.5848 ≈ 0.4152

Final Calculation

Now, we want to find PB, the probability that all three lines are in use:

PB p3 ≈ 0.41523 ≈ 0.071575

Therefore, the probability that all three telephone lines are in use at any given time is approximately 0.071575, or 7.1575%.

Conclusion

By following a step-by-step approach, we have calculated the probability that all three lines are in use given the probability that at least one line is in use. This method highlights the importance of understanding conditional probabilities and independence in statistical analysis. For office managers and IT professionals, such calculations can help optimize communication resources and ensure efficient use of available lines.