We describe new families of discrete distributions that are used to model sums of exchangeable Bernoulli random variables. These discrete distributions can be parameterized in terms of their range, mean, variance, and shape parameters. These models are fitted to an example involving mortality rates for children in a survey of families in Brazil. The methods illustrate that mortality rates in this survey increase with family size and that the correlation of within-family mortality status also depends on the family size. These methods are also applied to a laboratory study of birth defects in mice.