cannot determine the size of a 1000-year event based on such records alone but instead must use a statistical model to predict the magnitude of such an (unobserved) event. Even if the historic return interval is a lot less than 1000 years, if there are a number of less-severe events of a similar nature recorded, the use of such a model is likely to provide useful information to help estimate the future return interval. Probability distributions One would like to be able to interpret the return period in probabilistic models. The most logical interpretation for this is to take the return period as the counting rate in a Poisson distribution since it is the expectation value of the rate of occurrences. An alternative interpretation is to take it as the probability for a yearly Bernoulli trial in the binomial distribution. That is disfavoured because each year does not represent an independent Bernoulli trial but is an arbitrary measure of time. This question is mainly academic as the results obtained will be similar under both the Poisson and binomial interpretations. Poisson The probability mass function of the Poisson distribution is where is the number of occurrences the probability is calculated for, the time period of interest, is the return period and is the counting rate. The probability of no-occurrence can be obtained simply considering the case for . The formula is Consequently, the probability of exceedance (i.e. the probability of an event "stronger" than the event with return period to occur at least once within the time period of interest) is Note that for any event with return period , the probability of exceedance within an interval equal to the return period (i.e. ) is independent from the return period and it is equal to . This means, for example, that there is a 63.2% probability of a flood larger than the 50-year return flood to occur within any period of 50 year. Example If the return period of occurrence is 234 years () then the probability of exactly one occurrence in ten years is Binomial In a given period of n years, the probability of a given number r of events of a return period is given by the binomial distribution as follows. This is valid only if the probability of more than one occurrence per year is zero. Often that is a close approximation, in which case the probabilities