- Mean calls per day: \(\lambda = \frac{300}{100} = \boxed{3}\).
- Probability of exactly 4 calls: \(P(X = 4) = \frac{3^4 \cdot e^{-3}}{4!} \approx \boxed{0.168}\).
- Expected number of days with 4 calls: \(100 \cdot P(X=4) \approx \boxed{16.8}\).
Explanation
- Calculate the mean calls per day
We are given that a business observed a total of \(300\) incoming support calls over a period of \(100\) days. To find the mean number of calls per day (\(\lambda\)), we divide the total number of calls by the total number of days:
\[\lambda = \frac{\text{Total Calls}}{\text{Total Days}} = \frac{300}{100} = 3\]
So, on average, the business receives \(3\) calls per day.
- Calculate the probability of exactly 4 calls using the Poisson model
Under the assumption that the number of calls per day follows a Poisson distribution with mean \(\lambda = 3\), we use the Poisson probability mass function to find the probability of receiving exactly \(k=4\) calls in a single day:
\[P(X = k) = \frac{\lambda^k e^{-\lambda}}{k!}\]
Substituting \(\lambda = 3\) and \(k = 4\):
\[P(X = 4) = \frac{3^4 \cdot e^{-3}}{4!} = \frac{81 \cdot e^{-3}}{24} = \frac{27}{8} \cdot e^{-3}\]
Using the value of \(e^{-3} \approx 0.049787\):
\[P(X = 4) \approx \frac{81 \cdot 0.049787}{24} \approx 0.1680\]
The probability of having exactly \(4\) calls on a given day is approximately \(0.1680\).
- Calculate the expected number of days with 4 calls
To find the expected number of days out of \(100\) that have exactly \(4\) calls, we multiply the total number of days by the probability of having \(4\) calls on any given day:
\[\text{Expected Days} = 100 \cdot P(X = 4) \approx 100 \cdot 0.1680 = 16.80\]
Therefore, we would expect approximately \(16.8\) days out of \(100\) to have exactly \(4\) calls.
Examples
This Poisson model is highly practical for businesses to predict workload. For instance, if an IT department knows the average number of daily tickets, they can use the Poisson distribution to estimate the probability of high-demand days (like receiving 4 or more calls) and schedule their staff accordingly to maintain efficient service. In this case, by calculating that they can expect about \(16.8\) days with exactly \(4\) calls, the manager can better allocate resources to handle these specific peak periods.