The Poisson formula is a mathematical expression used in probability theory and statistics to describe the distribution of events occurring within a fixed interval of time or space. It is particularly useful for modeling rare events, such as the number of phone calls received at a call center in an hour or the number of accidents at a traffic intersection over a day. The formula is based on the Poisson distribution, which is characterized by a single parameter, λ (lambda), representing the average rate of occurrence. In its simplest form, the Poisson formula calculates the probability of observing a specific number of events (k) given the average rate (λ). The formula is expressed as P(X = k) = (λ^k * e^(-λ)) / k!, where e is the base of the natural logarithm, and k! is the factorial of k. This formula helps researchers and analysts understand and predict the likelihood of various outcomes in fields such as queueing theory, {telecommunications