How does the Poisson distribution model rare events over many short time intervals?

The Poisson distribution models rare events by treating them as a series of independent trials where the probability of success in any single, tiny interval is extremely low.12

Mathematically, this is derived by dividing a period of time or space into a large number ($n$) of very small, equal intervals. If the probability of an event occurring in any one interval ($p$) is inversely proportional to $n$, the expected number of occurrences ($\lambda =np$) remains constant even as the number of intervals approaches infinity.12

Under these conditions, the probability of two or more events occurring within a single interval becomes negligible. As the interval length shrinks toward zero, the binomial distribution converges to the Poisson distribution, defined by the probability mass function:2

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

In this model, $\lambda$ represents both the mean and the variance of the distribution. This framework is widely used to analyze phenomena such as radioactive decay or the arrival of calls at a facility.123