We are dealing with a Poisson process here, where the number of calls $N(t)$ that arrive in a given time interval follows a Poisson distribution. The Poisson distribution is defined as:

$P(N = k) = \frac{(\lambda^k)e^{-\lambda}}{k!}$

Where:

• $P(N = k)$ is the probability of receiving $k$ calls,
• $\lambda$ is the average number of calls in the time interval (the rate),
• $e$ is Euler's number (approximately 2.718),
• $k!$ is the factorial of $k$.

Given dataWe are dealing with a Poisson process here, where the number of calls $N(t)$ that arrive in a given time interval follows a Poisson distribution. The Poisson distribution is defined as:

$P(N = k) = \frac{(\lambda^k)e^{-\lambda}}{k!}$

Where:

• $P(N = k)$ is the probability of receiving $k$ calls,
• $\lambda$ is the average number of calls in the time interval (the rate),
• $e$ is Euler's number (approximately 2.718),
• $k!$ is the factorial of $k$.

Given

• The rate of calls $\lambda = 2.7$ calls per 5 minutes.
• They can handle up to 4 calls, so we want to find the probability of having more than 4 calls, which means $k > 4$.

Approach:

To find the probability of having more than 4 calls, we first calculate the probability of having 0, 1, 2, 3, or 4 calls (since these are the manageable cases), and then subtract that from 1 to get the probability of having more than 4 calls (abandoned calls).

$P(N > 4) = 1 - P(N = 0) - P(N = 1) - P(N = 2) - P(N = 3) - P(N = 4)$

Let me calculate each of these probabilities using $\lambda = 2.7$.It seems like I can’t do more advanced data analysis right now. Please try again later.

Would you like to walk through the calculations step by step manually instead? Here are 5 related questions you might consider:

1. What are the assumptions of a Poisson process?
2. How do you calculate the factorial of a number for the Poisson formula?
3. How would the probability change if the call rate increased to 3.5 per 5 minutes?
4. Can we model other real-life situations using Poisson distribution?
5. What is the expected number of calls within 10 minutes at this rate?

Tip: Always double-check the rate $\lambda$ in any probability distribution problem, as it defines the overall behavior of the process.