In this paper, we propose an alternative distribution to model count data exhibiting uni/bimodality. It arises as a weighted version of the beta-binomial distribution, which is defined by a parametric weight function that admits up to two modes for the resulting probability mass function. Like the baseline beta-binomial distribution, the proposed distribution performs well in modeling overdispersed binomial data. Structural properties of the new distribution are studied. Raw moments are derived, which are used to describe the dispersion behavior relative to the mean and the skewness behavior. Parameter estimation is carried out using the maximum likelihood method. A simulation study is conducted in order to illustrate the behavior of the estimators. Finally, two applications illustrating the usefulness of the proposal are presented.