The Poisson probability formula, represented as $$p(x=k) = \frac{e^{-\lambda} \lambda^{k}}{k!}$$, calculates the probability of a given number of events happening in a fixed interval of time or space, given that these events occur with a known constant mean rate \(\lambda\) and independently of the time since the last event. This formula is crucial in understanding how often certain events will happen under specific conditions, such as customer arrivals or system failures, making it a powerful tool for modeling random processes in various fields.
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