Restriction sites or other sequence patterns are usually assumed to occur according to a Poisson distribution with mean equal to the reciprocal of the probability of the given site or pattern. For situations where non-overlapping occurrences of patterns, such as restriction sites, are the objects of interest, this note shows that the Poisson assumption is frequently misleading. Both the case of base composition (independent bases) and of dinucleotide frequencies (Markov chains) are treated. Moreover, a new technique is presented which allows treatment of collections of patterns, where the departure from the Poisson assumption is even more striking. This later case includes double digests, and an example of a five enzyme digest is included.