We introduce mixtures of binomial distributions derived by assuming that the probability parameter p varies according to some law. We use the transformation p = exp(-t) and consider various appropriate densities for the transformed variables. In the process, the Laplace transform becomes the fundamental entity. Large numbers of new binomial mixtures are generated in this way. Some transformations may involve several variates that lead to "multivariate" binomial mixtures. An extension of this to the logarithmic distribution, with parameter p, is possible. Frullani integrals and Laplace transforms are encountered.

Graphical representations of some of the more significant distributions are given. These include probability functions, regions of validity, and three dimensional representations of probability functions showing the response to variation of parameters when two parameters are involved.