Final answer:
To answer the student's question, the Poisson distribution formula is used to calculate the probabilities for the number of telephone lines in use at a given time. The Poisson distribution is suitable here because calls occur at a constant rate and independently of each other, making it simple to compute the probability for any number of calls.
Step-by-step explanation:
To calculate the probability that a business with six customer service telephone lines will be using a certain number of lines (denoted by x) at any given time, we can use the Poisson distribution. The Poisson distribution applies here because calls come in at a constant average rate, and the calls are independent of each other. Given a Poisson distribution with a mean (λ) which equals the average number of calls, the formula for calculating the probability of x calls is:
P(X = x) = (λ^x * e^-λ) / x!
Using this formula, we can calculate the probabilities for x = 0, 1, 2, 3, 4, 5, and 6. For example, if the average number of calls (λ) is 5.5, then the probability of receiving exactly 5 calls is:
P(X = 5) = (5.5^5 * e^-5.5) / 5!
To find the probability of receiving at most six calls, we would calculate the cumulative probability up to and including six calls:
P(x ≤ 6) = Σ from x=0 to 6 P(X = x)
We can use a calculator or software that has a Poisson cumulative distribution function (poissoncdf) to make these calculations easier.