Contagious statistical distributions are a valuable resource for managing contagion by means of k-connected chains of distributions. Binomial, hypergeometric, Pólya, uniform distributions with the same values for all parameters except sample size n are known to be strongly associated. This paper describes how the relationship can be obtained via factorial moments, simplifying the process by including novel elements. We describe the properties of these distributions and provide examples of their real-world application, and then define a chain of k-connected distributions, which generalises the relationship among samples of any size for a given population and the Pólya urn model.