Calls to a switchboard

Question

A switchboard receives on average $4$ calls per hour (Poisson). What is the probability of receiving exactly $2$ in one hour?

Solution

Number of rare events in a time interval: $X\sim\text{Pois}(\lambda)$ with $\lambda = 4$. $P(X=2) = e^{-\lambda}\frac{\lambda^2}{2!} = e^{-4}\cdot\frac{16}{2} = 8\,e^{-4} \approx 8\cdot 0{,}0183 = \boxed{0{,}147}.$ About $14{,}7\%$.

Links

Topics: Probability distributions
Concepts: Poisson distribution
Skills: Probability calculation · Modelling
People: Siméon-Denis Poisson
Exercise type: Probability problem