This paper presents a new departure in the generalization of the binomial distribution by adopting the assumption that the underlying Bernoulli trials take on the values α or β where α < β, rather than the conventional values 0 or 1. The adoption of this more general assumption renders the binomial distribution a four-parameter distribution of the form B(n,p,α,β), and requires the generalization of Romanovsky's (1923) reduction formula for central moments. This paper assesses the usefulness of B(n,p,α,β), and its reduction formula, in the numerical analysis of two problems of interest to decision theorists.