Phone calls arrive at the rate of 48 per hour at the reservation desk for Regional Airways. (Round your answers to four decimal places.)(a) Compute the probability of receiving two calls in a 5-minute interval of time.(b) Compute the probability of receiving exactly 10 calls in 15 minutes.(c) Suppose no calls are currently on hold. If the agent takes 5 minutes to complete the current call. how many callers do you expect to be waiting by that time?

(a) To find the probability of receiving two calls in a 5-minute interval of time, we need to first convert the arrival rate to a rate per 5 minutes. There are 12 five-minute intervals in an hour, so the arrival rate per 5 minutes is:

λ = (48 calls/hour) / (12 intervals/hour) = 4 calls/5 minutes

Using the Poisson distribution with parameter λ = 4, we can calculate the probability of receiving exactly 2 calls in a 5-minute interval:

P(X = 2) = (e^(-λ) * λ^2) / 2! = (e^(-4) * 4^2) / 2! ≈ 0.1465

Therefore, the probability of receiving two calls in a 5-minute interval is approximately 0.1465.

(b) To find the probability of receiving exactly 10 calls in 15 minutes, we need to first convert the arrival rate to a rate per 15 minutes. There are 4 fifteen-minute intervals in an hour, so the arrival rate per 15 minutes is:

λ = (48 calls/hour) / (4 intervals/hour) = 12 calls/15 minutes

Using the Poisson distribution with parameter λ = 12, we can calculate the probability of receiving exactly 10 calls in a 15-minute interval:

P(X = 10) = (e^(-λ) * λ^10) / 10! = (e^(-12) * 12^10) / 10! ≈ 0.1032

Therefore, the probability of receiving exactly 10 calls in 15 minutes is approximately 0.1032.

(c) The expected number of callers waiting by the time the agent completes the current call can be found using the formula:

E(N) = λ * t

where λ is the arrival rate and t is the time the agent takes to complete the call. Since λ = 48 calls/hour and the agent takes 5 minutes to complete the call, we need to convert the arrival rate to a rate per 5 minutes:

λ = (48 calls/hour) / (60 minutes/hour) * 5 minutes = 4 calls/5 minutes

Then, we can calculate the expected number of callers waiting:

E(N) = λ * t = 4 * 5 = 20

Therefore, we expect there to be 20 callers waiting by the time the agent completes the current call.

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