The Poisson distribution describes the probability of a number of independent events occurring in a fixed interval of time or space when the average rate of occurrences is known. It can be used when there are a large number of trials with a small probability of success, like the number of deaths from horse kicks in the Army each year. The mean number of successes is equal to the number of trials multiplied by the probability of success. The Poisson distribution becomes applicable when the number of trials approaches infinity and the probability of success approaches zero while the mean remains constant. It is used to model random events that are expected to occur with a consistent average rate.